3.1: Introduction to Probability
Learning Objectives
- Learn key initial terminology about probability
- Determine the sample space of a given situation
- Recognize and use the three basic methods of determining probability measures
- Explain the importance of the Law of Large Numbers
- Describe the complement of an event
- Use the Complement Probability Rule for determining the probability of an event
Review and Preview
Inferential statistics seeks to make educated guesses about populations using statistics from randomly chosen samples. The usefulness of a sample statistic depends on the sample from which it was taken. We cannot guarantee that samples are representative of the population, but we can ensure that any bias present is due to random chance. What is the likelihood that our randomly chosen sample is representative? In other words, what is the probability that our sample statistic is close to the population parameter? These are fundamental questions to examine through the science of probability.
We hear probability claims daily. The weather forecast states a \(30\%\) chance of rain. The probability of a faulty product coming off a manufacturing line is \(6.5\%\). One has a \(20\%\) probability of purely guessing an answer correctly on a single multiple-choice question. A cancer research group believes that \(40\%\) of women and \(45\%\) of men will have a diagnosis of some type of cancer during their lifetimes. (Note: This means that if we randomly select a man, we have a \(45\%\) chance of choosing someone who has had or will have a cancer diagnosis in his life. This does not mean that any specific man has a \(45\%\) chance of being diagnosed with cancer in his lifetime.)
Each of the above scenarios involves a situation in which something will happen, and an outcome will occur, but we are uncertain which outcome it will be. Will it rain, or will it not rain? Is the next product produced of high quality or not? In statistics, we refer to such situations as random experiments. We have a clear context and an idea of possible outcomes, one of which will happen, and then use probabilities to measure the likelihood of any particular outcome.
As we move through the course, we will develop an understanding of random sampling to build probabilities. We will develop the ability to measure how unusual a sample statistic is within a given situation. If the probability of a calculated sample statistic is "small," then we conclude that the outcome is unusual. The use of probability goes well beyond this application; therefore, we develop probability within a larger context.
Basic Concept of Probability
We begin by examining the meaning of the term probability. Generally, probability is a numerical value inclusively between \(0\) and \(1 \) measuring the likelihood that a specific event will occur within a given situation. The representation of that numerical value might be in decimal form, fractional form, or percentage form, such as \(0.75\) \(=\frac{3}{4}\) \(=75\%.\) An outcome of a random experiment is a potential result of the experiment. The term event is a set of outcomes one might expect from a given random experiment. For example, if one rolls a pair of standard game dice as shown in Figure \(\PageIndex{1}\), a possible event could be both dice landing with one facing up.
Figure \(\PageIndex{1}\): Game Dice
Before measuring probability accurately, we must clearly describe the given situation (such as rolling a pair of standard game dice) and the event of interest (such as both dice landing with one facing up). For a given clearly described situation, the collection of all possible outcomes is called the
sample space
or
event space
. For reasonably simple situations, one can fully describe the sample space.
What is the sample space if one rolls a pair of standard game dice, as shown in Figure \( \PageIndex{1} \)?
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The graphic below shows the sample space, where one die outcome is red and the other in white. Notice there are thirty-six possible outcomes. We should also notice that, in general, we must be concerned with the order of the two dice.  Why is this? Because these are two outcomes that we can distinguish. Since probability measures the degree of uncertainty, one must consider all available information. We produce a complete sample space by carefully reflecting on all possible outcomes.
Figure \(\PageIndex{2}\): Sample space of rolling a pair of standard game dice
When rolling a pair of dice, the event of rolling two ones is an outcome. However, the event in which both dice land with a sum of five describes several outcomes in the sample space: \(1\) and \(4;\) \(2\) and \(3;\) \(3\) and \(2;\) and \(4\) and \(1.\)
Once we understand the sample space, we can examine the probability measures of various events within that situation. A probability measure near \(1\) indicates that the specific event is more likely to occur, and a probability measure near \(0\) indicates that the specific event is less likely to occur. We use the symbols \(P \left( A \right) \) to indicate the probability of a specific event \( A.\) For example, \(P \left( \text{rain} \right)\) \(=.30\) \(= 30\% \) indicates "the probability of rain is \(30\%\)". If for some event \(A,\) we find that \(P \left( A \right)\) \(=1.00\) \(= 100\%, \) then we say this event is a certain event . Similarly, if for some event \(A,\) we find that \(P \left( A \right)\) \(=0.00\) \(= 0\%,\) then we say this event is an impossible event . For finite sample spaces, impossible events cannot occur, and certain events must occur. For infinite sample spaces, the situation is more complicated; we will discuss this further in chapter \(4.\) For now, we are usually interested in events that are possible but not likely to occur.
In inferential statistics, we are often interested in outcomes that are not likely to happen; that is, they are not very probable; they are unusual. When improbable outcomes occur, either something rare happens, or we have reason to think our understanding of the situation needs to be updated, but what is the cutoff between improbable and probable, between unusual and usual? There is no fixed, agreed-upon value that will work for every situation. For this reason, we will often need to set a probability measure for considering an event improbable. Different individuals can generally feel the same event usual/likely or unusual/unlikely, depending on their personal life experiences. For example, if the probability of rain is \(15\%,\) one person may consider rain unlikely, and another may think it reasonably likely. A commonly chosen boundary measure for many statistical analysis areas is \(5\%.\) We will initially require a probability measure of \(5\%\) or less to label an event as unusual. Later, we will relax on this requirement, even leaving the choice to the consumer of our statistical measures to apply their chosen probability measure for unusual.
- Explain why \(150\%\) cannot be the probability of some event.
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The value \(150\%\) is larger than \(100\%.\) Any valid probability value will be between \(0.00\) \(= 0\%\) and \(1.00\) \(= 100\%,\) inclusively. Knowing such will help us recognize improper probabilities.
- Explain why \(1.21\) cannot be the probability of some event.
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The value \(1.21\) is larger than \(1.00.\) Again, any valid probability value will be between \(0.00\) \(= 0\%\) and \(1.00\) \(= 100\%,\) inclusively.
- Explain why \(-0.22\) cannot be the probability of some event.
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The value \(-0.22\) is negative. Any valid probability value will be non-negative since the value must be between \(0.00\) \(= 0\%\) and \(1.00\) \(= 100\%.\)
- Give an example of a situation and an impossible event for that situation.
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Answers to this can vary greatly. An example tied to the situation about rolling pairs of dice would be the event of rolling two standard game dice with a sum of \(13\) up. Since the largest each die can be is \(6,\) the sum cannot exceed \(12;\) the event is an impossible event. That is, \(P \left( \text{rolling a sum of }13 \right)\) \(= 0.00\%.\)
There is one final fundamental property of probability for the events and sample space within any given situation. The sum of the probabilities of all possible outcomes within the sample space of a given situation must always total \( 1.00\) \(= 100\%.\)
Basic Methods for Computing Simple Probabilities
With our basic terminology established, we focus on how to compute a given situation's probability. We initially examine three basic and commonly used methods. Our first method is called the subjective or intuitive method, where we produce a numerical estimate of the probability based on personal judgment, past experiences, or even personal opinion. For example, having purchased a few non-winning lottery tickets in the past, yet hearing of a winner on the news, we might estimate the probability of winning as a low value of \(1\%.\) We do not depend on subjectively determined probability values in quality statistical work. Instead, we form more robust and accurate methods for determining these values.
This leads us to our second method, commonly called the classical method. In this method, if a given situation has \(n\) different equally likely outcomes in the sample space and if event \(A \) can occur in \(x\) ways, then \[ \nonumber P\left( A \right) = \frac{\text{number of ways event }A \text{ can occur}}{\text{number of outcomes in the sample space}} = \frac{x}{n}. \]As an example, if a standard die is fair (so each face of the die has equal chance of landing up when the die is rolled), then \[ P\left( \small{\text{ROLLING A THREE}}\normalsize \right) = \frac{\text{number of ways rolling a three can occur}}{\text{number of outcomes in the sample space of rolling a die}} = \frac{1}{6} \approx 0.1667 = 16.67\% \nonumber\]Notice that this outcome would not be considered unusual since the probability measure is over \(5\%.\)
Consider the situation in which a pair of fair dice are rolled. Find the probability of each event given. Write results in probability notation and determine if the outcome is considered unusual.
- \(\small{\text{ROLLING }}\)\(\small{\text{A }}\)\(\small{\text{THREE }}\)\(\small{\text{ON }}\)\(\small{\text{THE }}\)\(\small{\text{FIRST }}\)\(\small{\text{DIE }}\)\(\small{\text{AND }}\)\(\small{\text{A }}\)\(\small{\text{FOUR }}\)\(\small{\text{ON }}\)\(\small{\text{THE }}\)\(\small{\text{SECOND }}\)\(\small{\text{DIE}}\).
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Figure \(\PageIndex{3}\): Sample space of rolling a pair of standard game dice
Since the dice are said to be fair dice, each of the thirty-six outcomes shown in our sample space above is equally likely. We will use the classical approach to determining the probability, using \( A \) to represent our event of \(\small{\text{ROLLING }}\)\(\small{\text{A }}\)\(\small{\text{THREE }}\)\(\small{\text{ON }}\)\(\small{\text{THE }}\)\(\small{\text{FIRST }}\)\(\small{\text{DIE }}\)\(\small{\text{AND }}\)\(\small{\text{A }}\)\(\small{\text{FOUR }}\)\(\small{\text{ON }}\)\(\small{\text{THE }}\)\(\small{\text{SECOND }}\)\(\small{\text{DIE}}\). We notice only one outcome in the sample space matches the event description. Therefore, \( P\left( A \right)\) \(= \frac{1}{36}\) \(\approx 0.0278\) \(= 2.78\%.\) Since this probability measure is less than \(5\%,\) then for our course, we will consider this outcome as unusual.
- \(\small{\text{ROLLING }}\)\(\small{\text{A }}\)\(\small{\text{THREE }}\)\(\small{\text{ON }}\)\(\small{\text{ONE }}\)\(\small{\text{DIE }}\)\(\small{\text{AND }}\)\(\small{\text{A }}\)\(\small{\text{FOUR }}\)\(\small{\text{ON }}\)\(\small{\text{THE }}\)\(\small{\text{OTHER }}\)\(\small{\text{DIE}}\).
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Figure \(\PageIndex{4}\): Sample space of rolling a pair of standard game dice
Using similar reasoning as above and using \( B \) to represent our event of \(\small{\text{ROLLING }}\)\(\small{\text{A }}\)\(\small{\text{THREE }}\)\(\small{\text{ON }}\)\(\small{\text{ONE }}\)\(\small{\text{DIE }}\)\(\small{\text{AND }}\)\(\small{\text{A }}\)\(\small{\text{FOUR }}\)\(\small{\text{ON }}\)\(\small{\text{THE }}\)\(\small{\text{OTHER }}\)\(\small{\text{DIE}}\), notice that there are two outcomes in the sample space that match the event description. Therefore, \( P\left( B \right)\) \(= \frac{2}{36}\) \(=\frac{1}{18}\) \(\approx 0.0556\) \(= 5.56\%.\) Since this probability measure is more than \(5\%,\) then for our course, we will not consider this outcome as unusual.
- \(\small{\text{ROLLING }}\)\(\small{\text{TWO }}\)\(\small{\text{DICE }}\)\(\small{\text{IN }}\)\(\small{\text{WHICH }}\)\(\small{\text{THE }}\)\(\small{\text{SUM }}\)\(\small{\text{OF }}\)\(\small{\text{THE }}\)\(\small{\text{NUMBER }}\)\(\small{\text{IS }}\)\(\small{\text{SIX}}\).
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Figure \(\PageIndex{5}\): Sample space of rolling a pair of standard game dice
Let \( C \) represent our event of \(\small{\text{ROLLING }}\)\(\small{\text{TWO }}\)\(\small{\text{DICE }}\)\(\small{\text{IN }}\)\(\small{\text{WHICH }}\)\(\small{\text{THE }}\)\(\small{\text{SUM }}\)\(\small{\text{OF }}\)\(\small{\text{THE }}\)\(\small{\text{NUMBER }}\)\(\small{\text{IS }}\)\(\small{\text{SIX}}\), we notice that there are five outcomes in the sample space that match the event description...can you find all of these? Therefore, \( P\left( C \right)\) \(= \frac{5}{36}\) \(\approx 0.1389\) \(= 13.89\%.\) Since this probability measure is more than \(5\%,\) then for our course, we will not consider this outcome as unusual.
Our third method of computing probabilities is the empirical, experimental, or relative frequency method . In this method, we repeatedly conduct an experiment, noting the outcomes of the trials, to establish an estimate of probability measures. In repeating a given experiment \(n\) times, and noting event \(A\) occurred \(f\) times, then \[ \nonumber P\left( A \right) = \frac{\text{number of times event }A \text{ occurred}}{\text{number of times situation/experiment was repeated}} = \frac{f}{n}. \]Notice how this method relates to our previous work producing relative frequency distributions when summarizing data sets. In a sense, we were producing probability measures with our relative frequency values.
The quality of the probability estimate is dependent on the number of repeated trials used. For example, a researcher finds that \(25\) of \(150\) randomly selected Kansas teens texted while driving during the last week, empirically indicating that the proportion of Kansas teens that drove while texting last week is \( \frac{25}{150}\) \(\approx 16.67\%.\) Equivalently, there is approximately a \(16.67\%\) probability that a randomly selected Kansas teen drove while texting last week. This measure based on only \(150\) teens does not give us high confidence in this estimated probability value; if the researcher could collect data from \(1,500\) Kansas teens, we could be more confident in the estimation.
In general, we require the number of carefully designed repeated trials to be as large as possible to produce an estimate of a probability value. Two underlying assumptions must be used with the experimental frequency approach. First, if an event occurred with a certain probability in past trials, this same event will occur about the same percentage of times in future trials. Second, the relative frequency probability of an event will tend to approach the true probability value as more and more trials are measured (this is commonly referred to as the Law of Large Numbers ).
There are times when simulations (especially computer simulations) produce a large number of trials and a reasonably accurate measure of the true probability of specific events, especially when the outcomes in a situation are not equally likely. Much weather forecasting is based on computer simulation of outcomes based on regional weather conditions. Similarly, the spread of infectious diseases is modeled by computer simulations to predict outcomes while avoiding extensive medical research costs or without impacting living organisms.   Of course, only rolling five times is insufficient for us to have high confidence in the accuracy of the empirical probability measure.  This new empirical probability measure of \( P ( \small{\text{ROLL }}\)\(\small{\text{OF }}\)\(\small{\text{ONE}}\normalsize)\) \(= \frac{188}{300}\) \(\approx 62.67\% \) indicates that our die is not likely to be fair; the experimental probability for the one is not the expected fair value of \( P ( \small{\text{ROLL }}\)\(\small{\text{OF }}\)\(\small{\text{ONE}}\normalsize)\) \(= \frac{1}{6}\) \(\approx 16.67\%.\)
In \(1856,\) Gregor Mendel began to study different inherited features, such as the color of pea plants. According to one source , in a second-generation group of pea plants, \(6002\) peas produced by the plants were yellow, and \(2001\) were green in color. What was the empirical probability that a randomly selected second-generation pea would be green? Is this close to the hypothesized value that Mendel claimed of \( 25\%?\)
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We compute that \( P (\small{\text{GREEN }}\)\(\small{\text{COLORED }}\)\(\small{\text{PEA}}\normalsize)\) \(= \frac{2001}{6002 + 2001}\) \(\approx 25.0031\%.\) Mendel's experimental probability is extremely close to his hypothesized probability claim of \(25\%.\)
Complement and Complement Probabilities
There are times we will be interested in finding the probability that an event \(A\) does not occur. The collection of all outcomes in a sample space in which a given event \(A\) does not occur is called the complement of event \(A\) and is denoted by \(\bar{A}.\) Other sources may use different notations to denote the complement; common ones include \(A^c\) or \(\sim{A}.\) The idea of complementary events allows us to divide the sample space into two mutually exclusive groups (no outcome can be found in both of the two groups \(A\) and \(\bar{A}\)) and also exhaustive (every outcome of the sample space must be included in one of our two groups). For example, suppose \(A \) is the event of getting two different numbers on each die when two dice are rolled; there are \(30\) such outcomes in our sample space that meet this event description. \(\bar{A} \) is the event of getting the same numbers on each dice when two dice are rolled; there are \(6\) such outcomes in our sample space. Between \(A\) and \(\bar{A}\), all \(36\) outcomes appear exactly once.
Figure \(\PageIndex{2}\): Events \(A\) (blue background) and \(\bar{A}\) (grey background)
Since all outcomes of a situation must be either in the event or the complement of the event, we have the following three key consequences:
- \(P\left( A \right) + P\left( \bar{A} \right)\) \(= 1\) \(= 100\% \)
- \(1 - P\left( A \right)\) \(= P\left( \bar{A} \right) \)
- \(1- P\left( \bar{A} \right) = P\left( A \right)\)
Notice that the first consequence leads naturally to the other two with basic algebra. Suppose we wish to determine the probability of event \(A\) from above \((P ( \small{\text{GETTING }}\)\(\small{\text{TWO }}\)\(\small{\text{DIFFERENT }}\)\(\small{\text{NUMBERS }}\)\(\small{\text{OF }}\)\(\small{\text{PIPS }}\)\(\small{\text{ON }}\)\(\small{\text{EACH }}\)\(\small{\text{DIE }}\)\(\small{\text{WHEN }}\)\(\small{\text{TWO }}\)\(\small{\text{DICE }}\)\(\small{\text{ARE }}\)\(\small{\text{ROLLED}}\normalsize ))\). We can go back to our sample space to count all such outcomes or use the complement concept to produce our results quickly. We have\[ \begin{align*} P \left( A \right) &=1 - P \left( \bar{A} \right) \\[6pt] &= 1 - \frac{6}{36} \\[6pt] &= \frac{30}{36} = \frac{5}{6} \approx 0.8333 \text{ or } 83.33\% \end{align*}\]This example illustrates that it is sometimes easier to determine the probability of a complement event instead of the given event. Once we know the probability of the complement event, we can easily determine the probability of the event.
 The tiles are well-mixed. We will randomly draw one tile from the bag without looking into the bag. We want to determine the probability that we draw a tile that is not a primary color; that is, we are to find \( P ( \small{\text{NOT }}\)\(\small{\text{A }}\)\(\small{\text{PRIMARY }}\)\(\small{\text{COLORED }}\)\(\small{\text{TILE}}\normalsize ).\)
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As a reminder, there are three primary colors: red, yellow, and blue. Although the complement is not necessary to answer this question, the use of the complement makes the problem easier to compute. Since "a primary colored tile" is the complement event of "not a primary colored tile" in this situation, we notice that \( P ( \small{\text{NOT }}\)\(\small{\text{A }}\)\(\small{\text{PRIMARY }}\)\(\small{\text{COLORED }}\)\(\small{\text{TILE}}\normalsize )\) \(= 1 - P( \small{\text{A }}\)\(\small{\text{PRIMARY }}\)\(\small{\text{COLORED }}\)\(\small{\text{TILE}}\normalsize )\) \(= 1-\frac{6+9}{60}\) \(= \frac{45}{60}\) \(=75\%.\)
- A number is chosen randomly from the set of integers between \(1\) and \(99,\) inclusively. What is the probability of randomly selecting a number that is not a perfect square?
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We notice that there are only a few integers inclusively between \(1\) and \(99\) that are perfect squares and many that are not. Specifically the perfect square integers in this set are \( \{1,\) \(4,\) \(9,\) \(16,\) \(25,\) \(36,\) \(49,\) \(64,\) \(81\}.\) Thus \(P ( \small{\text{NOT }}\)\(\small{\text{A }}\)\(\small{\text{PERFECT }}\)\(\small{\text{SQUARE}}\normalsize)\) \(= 1 - P ( \small{\text{PERFECT }}\)\(\small{\text{SQUARE}}\normalsize)\) \(= 1 - \frac{9}{99}\) \(\approx 0.909\) \(= 90.9\%.\)
- Suppose we want to know the probability of randomly selecting a group of \(35\) people in which at least two people will have the same birth day in a year (such as September \(1^{st}\) or May \(28^{th}\) -- we will ignore leap years for simplicity). Suppose we also know the probability of randomly selecting a group of \(35\) people, in which no two people have the same birth day in the year, which is about \(18.56\%.\) (We will discuss how this value of \(18.56\%\) can be found later in the chapter). From this information, can we determine the probability of randomly selecting a group of \(35\) people in which at least two will have the same birth day in a year?
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We see that the complement of "at least two people will have the same birth day in a year" is the description "less than two people (that is none) will have the same birth day in a year." We can use our complement rule to note that \( P ( \small{\text{AT }}\)\(\small{\text{LEAST }}\)\(\small{\text{TWO }}\)\(\small{\text{WILL }}\)\(\small{\text{HAVE }}\)\(\small{\text{THE }}\)\(\small{\text{SAME }}\)\(\small{\text{BIRTHDAY}}\normalsize )\) \(= 1 - P( \small{\text{NO }}\)\(\small{\text{TWO }}\)\(\small{\text{WILL }}\)\(\small{\text{HAVE }}\)\(\small{\text{THE }}\)\(\small{\text{SAME }}\)\(\small{\text{BIRTHDAY}})\normalsize\) \(= 1-0.1856\) \(= 0.8144\) \(= 81.44.\%\)
There are times when clearly describing the complement of an event can be simple. For example, in the roll of a single standard game die, it is common to quickly describe the complement of \(\small{\text{AN }}\)\(\small{\text{EVEN }}\)\(\small{\text{NUMBER }}\)\(\small{\text{ON }}\)\(\small{\text{A }}\)\(\small{\text{ROLL}}\normalsize\) to be \(\small{\text{AN }}\)\(\small{\text{ODD }}\)\(\small{\text{NUMBER }}\)\(\small{\text{ON }}\)\(\small{\text{A }}\)\(\small{\text{ROLL}}\normalsize\). There are also times when clearly describing the complement of an event can be challenging. We must ensure the complement description covers all possible outcomes for a situation. For a different example, in random selection of a number from the real number line, it is common to quickly describe the complement of \(\small{\text{SELECTION }}\)\(\small{\text{OF }}\)\(\small{\text{A }}\)\(\small{\text{NEGATIVE }}\)\(\small{\text{NUMBER}}\normalsize\) as \(\small{\text{SELECTION }}\)\(\small{\text{OF }}\)\(\small{\text{A }}\)\(\small{\text{POSITIVE }}\)\(\small{\text{NUMBER}}\normalsize.\) However, there is the number, \(0,\) that is neither positive nor negative which means we have an incorrect complement description. The true complement to \(\small{\text{SELECTION }}\)\(\small{\text{OF }}\)\(\small{\text{A }}\)\(\small{\text{NEGATIVE }}\)\(\small{\text{NUMBER}}\normalsize\) is \(\small{\text{SELECTION }}\)\(\small{\text{OF }}\)\(\small{\text{A }}\)\(\small{\text{NON-NEGATIVE }}\)\(\small{\text{NUMBER}}\normalsize.\) Now all possible outcomes have been accounted for in the descriptions.
We must think carefully about our sample space and correctly identify all outcomes which are not in our event. As another example, we might have the situation of randomly selecting an FHSU student with a focus on the outcome of \(\small{\text{STUDENT }}\)\(\small{\text{HAS }}\)\(\small{\text{HAD }}\)\(\small{\text{AT }}\)\(\small{\text{LEAST }}\)\(\small{\text{THREE }}\)\(\small{\text{COVID }}\)\(\small{\text{VACCINE }}\)\(\small{\text{SHOTS}}\normalsize\). The complement is the outcome of \(\small{\text{STUDENT }}\)\(\small{\text{HAS }}\)\(\small{\text{HAD }}\)\(\small{\text{TWO }}\)\(\small{\text{OR }}\)\(\small{\text{FEWER }}\)\(\small{\text{COVID }}\)\(\small{\text{VACCINE }}\)\(\small{\text{SHOTS}}\normalsize\) or, equivalently, \(\small{\text{STUDENT }}\)\(\small{\text{HAS }}\)\(\small{\text{HAD }}\)\(\small{\text{FEWER }}\)\(\small{\text{THAN }}\)\(\small{\text{THREE }}\)\(\small{\text{COVID }}\)\(\small{\text{VACCINE }}\)\(\small{\text{SHOTS}}\normalsize\). Notice how the use of mathematical notation can help us here. We might represent the outcome of \(\small{\text{STUDENT }}\)\(\small{\text{HAS }}\)\(\small{\text{HAD }}\)\(\small{\text{AT }}\)\(\small{\text{LEAST }}\)\(\small{\text{THREE }}\)\(\small{\text{COVID }}\)\(\small{\text{VACCINE }}\)\(\small{\text{SHOTS}}\normalsize\) more briefly in notation as \(x \ge 3.\) Then, in the context of counting number of shots, the complement is \(x < 3,\) which leads to a proper complement description of \(\small{\text{STUDENT }}\)\(\small{\text{HAS }}\)\(\small{\text{HAD }}\)\(\small{\text{FEWER }}\)\(\small{\text{THAN }}\)\(\small{\text{THREE }}\)\(\small{\text{COVID }}\)\(\small{\text{VACCINE }}\)\(\small{\text{SHOTS}}\normalsize.\)
This leads to special cases that commonly cause problems in complement descriptions; specifically, events involving "at least," "at most," "all' or "none." The complement of "all are" is not "none are," but is instead "at least one is not." For example, the complement of \(\small{\text{STUDENT }}\)\(\small{\text{HAS }}\)\(\small{\text{HAD }}\)\(\small{\text{ALL }}\)\(\small{\text{AVAILABLE }}\)\(\small{\text{COVID }}\)\(\small{\text{VACCINE }}\)\(\small{\text{SHOTS}}\normalsize\) would be \(\small{\text{STUDENT }}\)\(\small{\text{HAS }}\)\(\small{\text{MISSED }}\)\(\small{\text{AT }}\)\(\small{\text{LEAST }}\)\(\small{\text{ONE }}\)\(\small{\text{COVID }}\)\(\small{\text{VACCINE }}\)\(\small{\text{SHOT}}\normalsize\). The complement of \(\small{\text{STUDENT }}\)\(\small{\text{HAS }}\)\(\small{\text{HAD }}\)\(\small{\text{NONE }}\)\(\small{\text{OF }}\)\(\small{\text{THE }}\)\(\small{\text{AVAILABLE }}\)\(\small{\text{COVID }}\)\(\small{\text{VACCINE }}\)\(\small{\text{SHOTS}}\normalsize\) would be \(\small{\text{STUDENT }}\)\(\small{\text{HAS }}\)\(\small{\text{HAD }}\)\(\small{\text{AT }}\)\(\small{\text{LEAST }}\)\(\small{\text{ONE }}\)\(\small{\text{OF }}\)\(\small{\text{THE }}\)\(\small{\text{AVAILABLE }}\)\(\small{\text{COVID }}\)\(\small{\text{VACCINE }}\)\(\small{\text{SHOTS}}\normalsize\). Note that the complement of "none are" is "at least one is." We must think carefully when dealing with complements of event claims involving such keywords.
Give written descriptions of the complements of each event described below
- The event: \(\small{\text{IT }}\)\(\small{\text{SNOWS }}\)\(\small{\text{ON }}\)\(\small{\text{CHRISTMAS }}\)\(\small{\text{DAY}}\normalsize\)
- Answer
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The complement event description would be \(\small{\text{IT }}\)\(\small{\text{DOES }}\)\(\small{\text{NOT }}\)\(\small{\text{SNOW }}\)\(\small{\text{ON }}\)\(\small{\text{CHRISTMAS }}\)\(\small{\text{DAY}}\normalsize\). We note that \(\small{\text{IT }}\)\(\small{\text{RAINS }}\)\(\small{\text{ON }}\)\(\small{\text{CHRISTMAS }}\)\(\small{\text{DAY}}\normalsize\) is an outcome that fits in the complement description, but is itself not the actual full complement since other outcomes exist in the complement besides rain.
- The event: \(\small{\text{NATASHA }}\)\(\small{\text{IS }}\)\(\small{\text{LESS }}\)\(\small{\text{THAN }}\normalsize 10 \small{\text{ MINUTES }}\)\(\small{\text{LATE}}\normalsize\)
- Answer
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Notice the given event description of less than \(10\) minutes can be represented mathematically as \(x < 10,\) informing us that the complement must be related to \(x \ge 10.\) Therefore, the complement event description would be \(\small{\text{NATASHA }}\)\(\small{\text{IS }}\)\(\small{\text{AT }}\)\(\small{\text{LEAST }}\normalsize 10 \small{\text{ MINUTES }}\)\(\small{\text{LATE}}\normalsize,\) or equivalently, \(\small{\text{NATASHA }}\)\(\small{\text{IS }}\)\(\small{\text{LATE }}\)\(\small{\text{BY }}\)\(\small{\text{AT }}\)\(\small{\text{LEAST }}\normalsize 10 \small{\text{ MINUTES}}\normalsize.\) The phrasing we might give can vary, but the meaning of the phrasing must be tied to the inequality \( x \ge 10.\)
- The event: \(\small{\text{ALL }}\)\(\small{\text{CARDS }}\)\(\small{\text{IN }}\)\(\small{\text{A }}\)\(\small{\text{POKER }}\)\(\small{\text{HAND }}\)\(\small{\text{ARE }}\)\(\small{\text{FACE }}\)\(\small{\text{CARDS}}\normalsize\) ( Click here for a full description and visualization of a standard deck of playing cards)
- Answer
-
The given event description of "all cards" is complemented by "at least one is not." So the complement event description would be \(\small{\text{AT }}\)\(\small{\text{LEAST }}\)\(\small{\text{ONE }}\)\(\small{\text{CARD }}\)\(\small{\text{IS }}\)\(\small{\text{NOT }}\)\(\small{\text{A }}\)\(\small{\text{FACE }}\)\(\small{\text{CARD}}\normalsize.\) We must think clearly about this complement and also note why the description \(\small{\text{NONE }}\)\(\small{\text{OF }}\)\(\small{\text{THE }}\)\(\small{\text{CARDS }}\)\(\small{\text{ARE }}\)\(\small{\text{FACE }}\)\(\small{\text{CARDS}}\normalsize\) is not the complement description. We could have one, two, three, or four cards that are not face cards as possible events in the complement.
- The event: \(\small{\text{NONE }}\)\(\small{\text{OF }}\)\(\small{\text{THE }}\)\(\small{\text{STUDENTS }}\)\(\small{\text{FAILED }}\)\(\small{\text{THE }}\)\(\small{\text{LAST }}\)\(\small{\text{EXAM}}\normalsize\)
- Answer
-
The given event description of "none failed" is complemented by "at least one did fail." So the complement event description would be \(\small{\text{AT }}\)\(\small{\text{LEAST }}\)\(\small{\text{ONE }}\)\(\small{\text{STUDENT }}\)\(\small{\text{FAILED }}\)\(\small{\text{THE }}\)\(\small{\text{LAST }}\)\(\small{\text{EXAM}}.\)
Summary
To review this section, we list several important facts to remember when working with probabilities:
- An outcome of a random experiment is any potential result of the experiment. An event is a set of outcomes one might expect from a given random experiment. The sample space is the collection of all possible outcomes in a given situation.
- The probability of an event \(A\) is denoted by \(P \left( A \right)\) with the condition that \(0 \le P \left( A \right) \le 1.\) If \(P \left( A \right) \le 5\%,\) we will currently consider the event as unusual.
- The sum of the probabilities for all the possible outcomes in the sample space will always total to \( 1\) \(= 100\%.\)
- Multiple methods may be used for computing probability values. We have discussed the methods of "subjective/intuition," "classical for equally likely simple events," and "empirical/experimental/relative frequency." Some methods only produce estimations of the actual probability value.
- The complement of an event \(A\) is denoted by \(\bar{A}\) and must contain all outcomes of the sample space that are not part of the given event. As such, we have the probability benefit of \( P \left( A \right) + P \left( \bar{A} \right) =1 \) that can be used to find the probability of an event if we know the probability of the complement.


9.6: Post-Traumatic Stress Disorder
Post-traumatic stress disorder (PTSD) is diagnosed in individuals who have been exposed to a traumatic event with chronic stress symptoms lasting more than one month that are so severe they interfere with relationships, school, or work. PTSD was formerly classified as an anxiety disorder but was placed in a new diagnostic category in the DSM-5 called “Trauma and Stressor-Related Disorders.”
Post-traumatic stress disorder has similar characteristics to severe anxiety and phobia-related disorders because of the physiological stress response that occurs. Post-traumatic stress disorder (PTSD) can develop in some people who have experienced a shocking, frightening, or dangerous event. It is natural to feel afraid during and after a traumatic situation, and the “fight-or-flight” stress response is a physiological reaction intended to protect a person from harm. Most people recover from the range of reactions that can occur after experiencing trauma. However, people who do not recover from these reactions and continue to experience problems are diagnosed with PTSD. People who have PTSD may feel stressed or frightened, even when they are not in danger. [1]
Symptoms
Symptoms of PTSD typically begin three months of the traumatic incident, but they may also begin years afterward. If symptoms occur within one month of the traumatic event, it is diagnosed as acute stress disorder. Symptoms must last more than a month and be severe enough to interfere with social or occupational functioning to be considered PTSD. The course of the illness varies; some people recover within six months, while others have symptoms that last much longer. In some people, the condition becomes chronic. [2] , [3]
To be diagnosed with PTSD, an adult must have the following types of symptoms for at least one month [4] , [5] :
- At least one “re-experiencing” symptom
- At least one “avoidance” symptom
- At least two “arousal and reactivity” symptoms
- At least two “cognition and mood” symptoms
Re-Experiencing Symptoms
Re-experiencing symptoms include the following [6] :
- Flashbacks—reliving the trauma over and over, including physical symptoms like a racing heart or sweating
- Bad dreams
- Frightening thoughts
Re-experiencing symptoms can start from the person’s own thoughts and feelings. Words, objects, or situations that are reminders of the event can also trigger re-experiencing symptoms. Re-experiencing symptoms may cause problems in a person’s everyday routine and relationships. [7]
Avoidance Symptoms
Avoidance symptoms are as follows [8] :
- Staying away from places, events, or objects that are reminders of the traumatic experience
- Avoiding thoughts or feelings related to the traumatic event
These symptoms may cause a person to change their personal routine. For example, after a car accident, a person who usually drives may avoid driving or riding in a car.
Arousal and Reactivity Symptoms
Arousal and reactivity symptoms include the following [9] :
- Being easily startled
- Feeling tense or “on edge”
- Having difficulty sleeping
- Having angry outbursts
Arousal symptoms are usually constant instead of being triggered by things that remind one of the traumatic events. These symptoms can make the person feel stressed and angry and can make it hard to do daily tasks, such as sleeping, eating, or concentrating. [10]
Cognition and Mood Symptoms
Cognition and mood symptoms are as follows [11] :
- Trouble remembering key features of the traumatic event
- Negative thoughts about oneself or the world
- Distorted feelings like guilt or blame
- Loss of interest in enjoyable activities
Cognition and mood symptoms can begin or worsen after the traumatic event and make the person feel alienated or detached from friends or family members. [12]
It is natural to have some of these types of symptoms for a few weeks after a traumatic event. However, when the symptoms last more than a month, seriously affect a person’s functioning, and are not related to substance use, medical illness, or anything except the event itself, they can be symptoms of PTSD. PTSD is also often accompanied by depression, substance abuse, or other anxiety disorders. [13]
Life Span Considerations
Children and teens can have extreme reactions to trauma, but they may exhibit different symptoms than adults. Symptoms of PTSD can be seen in young children (less than six years old) and may include the following:
- Bedwetting after having learned to use the toilet
- Forgetting how to talk or being unable to talk (i.e., selective mutism)
- Acting out the scary event during playtime
- Being unusually clingy with a parent or other adult
Older children and teens are more likely to show symptoms similar to those seen in adults. They may also develop disruptive, disrespectful, or destructive behaviors. Hypersexual behavior may occur if the trauma was related to a sexual assault. Older children and teens may also feel guilty for not preventing injury or death in certain traumatic situations and may have thoughts of revenge. [14]
Risk and Resilience Factors
Anyone can develop PTSD at any age, including war veterans; children who have experienced trauma; or adults who have experienced a physical or sexual assault, abuse, accident, disaster, being a refugee, or some other serious event. According to the National Center for PTSD, about 6 out of every 100 people will experience PTSD symptoms at some point in their lives. Women are more likely to develop PTSD than men, and genes may make some people more likely to develop PTSD than others. [15]
Not everyone with PTSD has directly experienced a dangerous event. Some people develop PTSD after a friend or family member experiences danger or harm. The sudden, unexpected death of a loved one can also lead to PTSD. [16] PTSD is triggered by events that are perceived to be life-threatening, and this can vary from individual to individual. It disrupts the general sense of safety that allows individuals to function in the world.
It is important to remember that not everyone who lives through a dangerous event develops PTSD. In fact, most people will not develop the disorder. Many factors play a part in whether a person will develop PTSD. Risk factors make a person more likely to develop PTSD, but other factors, called resilience factors, can help reduce the risk of developing the disorder or promote recovery from the disorder. [17]
These factors increase the risk for developing PTSD [18] :
- Living through dangerous events and traumas
- Being injured from a traumatic event
- Seeing another person hurt or seeing a dead body
- Experiencing childhood trauma or adverse childhood events (ACEs)
- Feeling horror, helplessness, or extreme fear
- Having little or no social support after the event
- Dealing with extra stress after the event, such as loss of a loved one, pain and injury, or loss of a job or home
- Having a history of mental illness or substance abuse
Review information about adverse childhood events (ACEs) in the “ Mental Health and Mental Illness ” section of Chapter 1.
Resilience factors that may promote recovery after trauma include the following [19] :
- Receiving support from other people, such as friends and family
- Finding a support group after a traumatic event
- Learning to feel good about one’s own actions in the face of danger, recognizing that we controlled what we could in an uncontrollable situation
- Having a positive coping strategy or a way of getting through the bad event and learning from it
- Being able to act and respond effectively despite feeling fear
If a child or adolescent discloses traumatic events to caregivers, teachers, or other adults, it is important for them to feel their concerns are validated by the adult in order to develop resilience.
Researchers are studying the importance of risk and resilience factors, as well as the impact of genetics and neurobiology. With more research, it may be possible to someday predict who is likely to develop PTSD and how to prevent it from occurring. [20] See Figure 9.8 [21] for an image of a veteran with PTSD using a service dog as an effective coping strategy.
Treatments
For people with PTSD, treatments include medications, psychotherapy, or a combination of both. Everyone is different, and PTSD affects people differently, so a treatment that works for one person may not work for another. It is important for anyone with PTSD to be treated by a mental health provider who has experience treating PTSD. [22]
If someone with PTSD is also experiencing an ongoing trauma, such as an abusive relationship, both of the problems need to be addressed. Other ongoing problems can include panic disorder, depression, substance use disorder, and suicidal ideation. [23]
Medications
Antidepressants can help control PTSD symptoms such as sadness, worry, anger, and feeling numb inside. For example, two FDA-approved medications for PTSD are sertraline and paroxetine. Other medications may be helpful for treating specific PTSD symptoms, such as sleep problems and nightmares. [24]
Psychotherapy
Psychotherapy can occur one-on-one or in a group setting. It typically lasts 6 to 12 weeks, but it can continue for as long as the individual finds it helpful. Research shows that additional support from family and friends can be an important part of recovery. [25]
Many types of psychotherapy can help people with PTSD. Some target the symptoms of PTSD directly, whereas other therapies focus on social, family, or job-related problems. Effective psychotherapies emphasize key components such as education about symptoms, identification of triggers or symptoms, and skills to manage the symptoms. Examples of psychotherapies used to treat PTSD are cognitive behavioral therapy, exposure therapy, eye movement desensitization and reprocessing, and animal therapy programs.
Cognitive behavioral therapy (CBT) combined with exposure therapy helps people face and control their fear by gradually exposing them to the trauma they experienced in a safe way. It uses imagining, writing, or visiting the place where the event happened to help reduce the intensity of PTSD symptoms. Read more about CBT in the “ Depressive Disorders ” chapter.
Cognitive restructuring helps individuals create new thought patterns about the trauma. Sometimes events are remembered differently than how they truly happened, and individuals may experience feelings of guilt or shame in relation to the trauma, regardless of whether they played an active role or not. They may feel guilt or shame about something that is not their fault. Therapists can help individuals assess the trauma through a variety of lenses to process and continue the healing journey.
Eye movement desensitization and reprocessing (EMDR) is a psychotherapy treatment that was originally designed to alleviate the distress associated with traumatic memories. During EMDR therapy, the client attends to emotionally disturbing material in brief sequential doses while simultaneously focusing on an external stimulus. Therapist-directed lateral eye movements are the most commonly used external stimulus, but a variety of other stimuli such as hand-tapping and audio stimulation are also used. It is hypothesized that EMDR therapy facilitates the individual’s access of their traumatic memory network so that new associations can be forged between the traumatic memory and more adaptive memories or information. These new associations are thought to result in elimination of emotional distress and development of cognitive insights. [26] , [27]
Animal assisted intervention (AAI), also referred to as animal therapy, is a commonly used complementary treatment for PTSD. It most often includes dogs or horses. A systematic review examined the outcomes in studies using AAI with trauma survivors. Although the reviewed studies were diverse and limited, all reported positive outcomes of AAI, such as reduced depression, PTSD symptoms, and anxiety. [28]
In 2017 a new therapy called methylenedioxymethamphetamine (MDMA) – assisted psychotherapy received Breakthrough Therapy Designation from the FDA for the treatment of PTSD. MDMA (also referred to by the illicit drug name Ecstasy) is administered under direct observation with psychotherapy in three monthly eight-hour sessions. Due to the limited number of sessions with direct observation, there is less chance of diversion, overdose, or withdrawal symptoms upon discontinuation. [29]
View the following YouTube video on PTSD [30] : NIMH-Funded Researcher Dr. Barbara Rothbaum Discusses Post-Traumatic Stress Disorder.
- National Institute of Mental Health. (2019, May). Post-traumatic stress disorder. U.S. Department of Health and Human Services. https://www.nimh.nih.gov/health/topics/post-traumatic-stress-disorder-ptsd ↵
- National Institute of Mental Health. (2019, May). Post-traumatic stress disorder. U.S. Department of Health and Human Services. https://www.nimh.nih.gov/health/topics/post-traumatic-stress-disorder-ptsd ↵
- American Psychiatric Association. (2013). Desk reference to the diagnostic criteria from DSM-5. ↵
- National Institute of Mental Health. (2019, May). Post-traumatic stress disorder. U.S. Department of Health and Human Services. https://www.nimh.nih.gov/health/topics/post-traumatic-stress-disorder-ptsd ↵
- American Psychiatric Association. (2013). Desk reference to the diagnostic criteria from DSM-5 . ↵
- National Institute of Mental Health. (2019, May). Post-traumatic stress disorder. U.S. Department of Health and Human Services. https://www.nimh.nih.gov/health/topics/post-traumatic-stress-disorder-ptsd ↵
- National Institute of Mental Health. (2019, May). Post-traumatic stress disorder. U.S. Department of Health and Human Services. https://www.nimh.nih.gov/health/topics/post-traumatic-stress-disorder-ptsd ↵
- National Institute of Mental Health. (2019, May). Post-traumatic stress disorder. U.S. Department of Health and Human Services. https://www.nimh.nih.gov/health/topics/post-traumatic-stress-disorder-ptsd ↵
- National Institute of Mental Health. (2019, May). Post-traumatic stress disorder. U.S. Department of Health and Human Services. https://www.nimh.nih.gov/health/topics/post-traumatic-stress-disorder-ptsd ↵
- National Institute of Mental Health. (2019, May). Post-traumatic stress disorder. U.S. Department of Health and Human Services. https://www.nimh.nih.gov/health/topics/post-traumatic-stress-disorder-ptsd ↵
- National Institute of Mental Health. (2019, May). Post-traumatic stress disorder. U.S. Department of Health and Human Services. https://www.nimh.nih.gov/health/topics/post-traumatic-stress-disorder-ptsd ↵
- National Institute of Mental Health. (2019, May). Post-traumatic stress disorder. U.S. Department of Health and Human Services. https://www.nimh.nih.gov/health/topics/post-traumatic-stress-disorder-ptsd ↵
- National Institute of Mental Health. (2019, May). Post-traumatic stress disorder. U.S. Department of Health and Human Services. https://www.nimh.nih.gov/health/topics/post-traumatic-stress-disorder-ptsd ↵
- National Institute of Mental Health. (2019, May). Post-traumatic stress disorder. U.S. Department of Health and Human Services. https://www.nimh.nih.gov/health/topics/post-traumatic-stress-disorder-ptsd ↵
- National Institute of Mental Health. (2019, May). Post-traumatic stress disorder. U.S. Department of Health and Human Services. https://www.nimh.nih.gov/health/topics/post-traumatic-stress-disorder-ptsd ↵
- National Institute of Mental Health. (2019, May). Post-traumatic stress disorder. U.S. Department of Health and Human Services. https://www.nimh.nih.gov/health/topics/post-traumatic-stress-disorder-ptsd ↵
- National Institute of Mental Health. (2019, May). Post-traumatic stress disorder. U.S. Department of Health and Human Services. https://www.nimh.nih.gov/health/topics/post-traumatic-stress-disorder-ptsd ↵
- National Institute of Mental Health. (2019, May). Post-traumatic stress disorder. U.S. Department of Health and Human Services. https://www.nimh.nih.gov/health/topics/post-traumatic-stress-disorder-ptsd ↵
- National Institute of Mental Health. (2019, May). Post-traumatic stress disorder. U.S. Department of Health and Human Services. https://www.nimh.nih.gov/health/topics/post-traumatic-stress-disorder-ptsd ↵
- National Institute of Mental Health. (2019, May). Post-traumatic stress disorder. U.S. Department of Health and Human Services. https://www.nimh.nih.gov/health/topics/post-traumatic-stress-disorder-ptsd ↵
- “ Local_veterans_group_gets_national_exposure_141127-F-ZZ999-001.jpg ” by unknown author is licensed in the Public Domain ↵
- National Institute of Mental Health. (2019, May). Post-traumatic stress disorder. U.S. Department of Health and Human Services. https://www.nimh.nih.gov/health/topics/post-traumatic-stress-disorder-ptsd ↵
- National Institute of Mental Health. (2019, May). Post-traumatic stress disorder. U.S. Department of Health and Human Services. https://www.nimh.nih.gov/health/topics/post-traumatic-stress-disorder-ptsd ↵
- National Institute of Mental Health. (2019, May). Post-traumatic stress disorder. U.S. Department of Health and Human Services. https://www.nimh.nih.gov/health/topics/post-traumatic-stress-disorder-ptsd ↵
- National Institute of Mental Health. (2019, May). Post-traumatic stress disorder. U.S. Department of Health and Human Services. https://www.nimh.nih.gov/health/topics/post-traumatic-stress-disorder-ptsd ↵
- EDMR Institute. (n.d.) What is EDMR? https://www.emdr.com/what-is-emdr/ ↵
- American Psychological Association. (2017, July 31). Clinical practice guidelines for the treatment of post-traumatic stress disorder: Eye Movement Desensitization and Reprocessing (EMDR) Therap y. https://www.apa.org/ptsd-guideline/treatments/eye-movement-reprocessing ↵
- O'Haire, M., Guerin, N., & Kirkham, A. (2015). Animal-Assisted Intervention for trauma: a systematic literature review. Frontiers in Psychology, 6 . https://doi.org/10.3389/fpsyg.2015.01121 ↵
- Feduccia, A. A., Jerome, L., Yazar-Klosinski, B., Emerson, A., Mithoefer, M. C., & Doblin, R. (2019). Breakthrough for trauma treatment: Safety and efficacy of MDMA-assisted psychotherapy compared to paroxetine and sertraline. Frontiers in Psychiatry, 10, 650. https://doi.org/10.3389/fpsyt.2019.00650 ↵
- National Institute of Mental Health (NIMH). (2021, June 21). NIMH-funded researcher Dr. Barbara Rothbaum discusses post-traumatic stress disorder [Video]. YouTube. All rights reserved. https://youtu.be/wIcWIbM4hLE ↵


6: Comparing Three or More Group Means
Hello everyone, and welcome to the sixth chapter of the University of Southern Queensland’s online, open access textbook for statistics for research students.
The aim of this chapter is to discuss the testing of differences between mean values on a central variable of interest between three or more groups. When we examined differences between two groups we used a form of the t-test. However, when moving to an examination of differences between three or more groups we can extend the logic of the t-test to a method known as Analyses of Variance or ANOVA.
There are some slides that appear via links within Chapter Six. Please look for these as you review the current chapter.


4.5: Theme and Variations
Building a piece of music around the presentation of a theme (a melodic idea) followed by a series of variations on that theme is not new to the Classical era. However, the frequent use of this structure in movements, often the 2nd movement, of larger Classical works such as symphonies and string quartets merits some additional study as we consider Classical forms.
In music, variation is a formal technique where material is repeated in an altered form. The changes may involve harmony, melody, counterpoint, rhythm, timbre, orchestration or any combination of these.
Variation Form
Variation forms include ground bass, passacaglia, chaconne, and theme and variations. Ground bass, passacaglia and chaconne are typically based on brief ostinato motifs providing a repetitive harmonic basis and are also typically continuous evolving structures. ‘Theme and variation’ forms are however based specifically on melodic variation, in which the fundamental musical idea, or theme, is repeated in altered form or accompanied in a different manner. ‘Theme and variation’ structure generally begins with a theme (which is itself sometimes preceded by an introduction), typically between eight and thirty-two bars in length; each variation, particularly in music of the eighteenth century and earlier, will be of the same length and structure as the theme. This form may in part have derived from the practical inventiveness of musicians; “Court dances were long; the tunes which accompanied them were short. Their repetition became intolerably wearisome, and inevitably led the player to indulge in extempore variation and ornament”; however, the format of the dance required these variations to maintain the same duration and shape of the tune.
Variation forms can be written as ‘free-standing’ pieces for solo instruments or ensembles, or can constitute a movement of a larger piece. Most jazz music is structured on a basic pattern of theme and variations.
Examples include John Bull’s Salvator Mundi , Bach’s Canonic Variations on “Vom Himmel hoch da komm’ ich her , Passacaglia and Fugue in C minor , Violin Chaconne , and (D minor solo violin suite), Corelli’s La Folia Variations , Beethoven’s Diabelli Variations , the Finale of Brahms’s Fourth Symphony, Variations on a Theme of Haydn , Op. 56, Elgar’s Enigma Variations , Franck’s Variations Symphoniques , and Richard Strauss’s Don Quixote . Both Schubert’s Death and the Maiden Quartet and Trout Quintet take their titles from his songs used as variation movements.
Contributors and Attributions
- Variation (music). Provided by : Wikipedia. Located at : http://en.Wikipedia.org/wiki/Variation_(music) . License : CC BY-SA: Attribution-ShareAlike


5: Blood Pressure Last updated Save as PDF Page ID 16792 Lapum et al. Ryerson University (Daphne Cockwell School of Nursing) via Ryerson University Library 5.1: Hypotension 5.2: Finding the Error Activity- Blood Pressure 5.3: Finding the Error Activity- Blood Pressure – Feedback 5.4: Try it Out 5.5: Try it Out- Two-step Blood Pressure 5.6: Try it Out- One-step Blood Pressure 5.7: Test Yourself 5.8: Test Yourself – Answers 5.9: Test Yourself- List in the Correct Order 5.10: Test Yourself- List in the Correct Order – Answers 5.11: How Best to View this Chapter? 5.12: Chapter Summary 5.13: What is Blood Pressure? 5.14: Why is Blood Pressure Measured? 5.15: Factors That Influence Blood Pressure 5.16: What are Blood Pressure Ranges? 5.17: How is Blood Pressure Measured? 5.18: Manual Blood Pressure Measurement 5.19: What Should the Healthcare Provider Consider? 5.20: Hypertension


4.4: The Middle Kingdom
Sculpture of the Middle Kingdom
Innovations during the Middle Kingdom included the solemnity evident in portraits of Senusret III and block statues.
Discuss the sculptural innovations during Egypt’s Middle Kingdom
Key Points
- Mentuhotep II was the first pharaoh of the Middle Kingdom , restoring stability after a period of pharaonic weakness and civil war. Senusret III is considered to be perhaps the most powerful Egyptian ruler of the dynasty and led the kingdom to an era of peace and prosperity.
- Senusret III is known for his strikingly somber sculptures in which he appears careworn and grave. While many statues portray him as a vigorous young man, others deviate from this standard and illustrate him as mature and aging. This is often interpreted as a portrayal of the burden of power and kingship.
- Another important innovation in sculpture during the Middle Kingdom was the block statue, which consisted of a man squatting with his knees drawn up to his chest.
- Aristocratic women’s role as art patrons during the Middle Kingdom speaks to their high status and gender equality.
Key Terms
- deified :To be treated as worthy of worship, or regarded as a god.
- Middle Kingdom :Egypt in the Twelfth and Thirteenth dynasties, between 2055 BC and 1650 BC.
Mentuhotep II
The Middle Kingdom (c. 2000-1650 BCE) was marked by the reunification of Egypt following a period of weak pharaonic power and civil war called the First Intermediate. Mentuhotep II restored stability in 2041 BCE after launching an attack that met with little resistance. After toppling the last rulers of the Tenth Dynasty, Mentuhotep II began consolidating his power over all Egypt, completing the process circa 2000 BCE. His subjects considered him to be divine or semi-divine, as suggested in a relief depicting the pharaoh receiving offerings .
Senusret III
During the Middle Kingdom, relief and portrait sculpture captured subtle, individual details that reached new heights of technical perfection. Some of the finest examples of sculpture during this time was at the height of the empire under Pharaoh Senusret III.
Senusret III (also written as Senwosret III or Sesostris III) ruled from 1878–1839 BCE and was the fifth monarch of the Twelfth Dynasty of the Middle Kingdom. His military campaigns gave rise to an era of peace and economic prosperity that not only reduced the power of regional rulers, but also led to a revival in craftwork, trade, and urban development in the Egyptian kingdom. One of the few kings who were deified and honored with a cult during their own lifetime, he is considered to be perhaps the most powerful Egyptian ruler of the dynasty.
Aside from his accomplishments in architecture and war, Senusret III is known for his strikingly somber sculptures in which he appears careworn and grave. Deviating from the standard way of representing kings, Senusret III and his successor Amenemhat III had themselves portrayed as mature, aging men. This is often interpreted as a portrayal of the burden of power and kingship. The change in representation as ideological, and not something to be interpreted as the portrayal of an aging king, is shown by the fact that in one single relief, Senusret III was represented as a vigorous young man, following the centuries old tradition, and as a mature aging king.
Block Statues and Women Patrons
Another important innovation in sculpture that occurred during the Middle Kingdom was the block statue, which would continue to be popular through to the Ptolemaic age almost 2,000 years later.  Often, these men are wearing a wide cloak that reduces the body of the figure to a simple block-like shape. In some cases the cloak covers the feet completely, and in others the feet are left uncovered. The head of the sculpture contains the most detail.
The sculpture pictured below—the fact that a private woman could have a sculpture made for herself—speaks volumes for the equality of gender in ancient Egypt. The heavy tripartite wig frames the broad face and passes behind the ears, thus giving the impression of forcing them forward. They are large in keeping with the ancient Egyptian ideal of beauty; the same ideal required small breasts, and in this respect, the sculpture is no exception. Whereas the natural curve of the eyebrows dips towards the root of the nose, the artificial eyebrows in low relief are absolutely straight above the inner corners of the eyes, a feature which places the bust early in the early Twelfth Dynasty. Around 1900 BCE, these artificial eyebrows, too, began to follow the natural curve and dipped toward the nose.
Tombs of the Middle Kingdom
Grand and royal tombs continued to be built for the deceased during the prosperous Middle Kingdom.
Compare and contrast the tombs and burial goods of the Middle Kingdom with those of the Old Kingdom.
Key Points
- As the pharaohs of the Middle Kingdom restored the country’s prosperity and stability, they stimulated a resurgence of art, literature, and monumental building projects, including tombs and burial goods .
- While burial goods continued to be common, objects of daily use were not typically included as they were in the Old Kingdom.
- Toward the end of the Middle Kingdom, new objects were introduced, such as shabtis and scarabs.
- Coffin Texts evolved from the previous Pyramid texts of the Old Kingdom, expanding and introducing spells that were more relatable to nobles and non-royal Egyptians.
Key Terms
- Pyramid Text :A collection of spells to protect the pharaoh from harm in the afterlife.
- Osiris :The god of the underworld and husband of the goddess Isis in the Egyptian pantheon. He is usually depicted with green skin and wrapped in a mummy’s linen.
- shabti :A figurine placed in tombs to act as servants for the deceased in the afterlife.
- sarcophagus :A stone coffin, often inscribed or decorated with sculpture.
- burial goods :Items included in tombs that were thought to be necessary in the afterlife, from everyday objects to jewelry and valuables.
Royal funerary practices in the Middle Kingdom remained much the same as in the Old Kingdom, with kings continuing to build pyramids for their burials. Unlike the Old Kingdom, however, Middle Kingdom royal pyramids were not quite as well constructed, and so few of them remain as pyramid structures today. Among the tombs built during this time are Amenemhat I’s funerary monument at El-Lisht; Sesostris I’s funerary monument; Amenemhat III’s pyramid at Hawara, which includes an elaborate labyrinth complex; and Sesostris II’s pyramid at Illahun. The construction of pyramids declined toward the end of the Twelfth Dynasty , as instability led to the decline of the Middle Kingdom.
Burial goods continued to be commonplace in tombs. Starting in the First Intermediate period, wooden models became very popular, often depicting everyday activities that the deceased expected to continue doing in the afterlife. The standard coffin was rectangular and brightly painted, often including an offering formula. Unlike the Old Kingdom, objects of daily use were not often included in the tombs; however, they reappeared toward the end of the Middle Kingdom.
Other new objects were introduced toward the end of the Middle Kingdom as well, including the first shabtis (also known as ushabtis) and the first scarabs. Shabtis were funerary figurines placed in tombs of the deceased to help them in the afterlife. Used from the Middle Kingdom until the end of the Ptolemaic Period nearly 2000 years later, most shabtis were of a small size, often covering the floor around a sarcophagus . Exceptional shabtis were of larger size, or produced as a one-of-a-kind master work. They were generally distinguished from other statuettes by being inscribed with the name of the deceased, his titles, and often with spells from the Coffin Texts . Shabtis were created to act as the deceased’s servants, performing the manual labor necessary for the plentiful existence of the afterlife.
Scarabs were popular amulets believed to be protectors of written products. The scarab was also used as a holder or medium for personal name seals. A figurine of a scarab would be carved out of stone, and then on the smooth stomach of the scarab, the engraving of a seal was made.
Another change in funerary practice during this time had to do with non-royal Egyptians. In contrast to elitist Old Kingdom attitudes towards the gods, the Middle Kingdom experienced an increase in expressions of personal piety and what could be called a democratization of the afterlife. In this worldview , all people possessed a soul and could be welcomed into the company of the gods after death.
In the Old Kingdom, the Pyramid Texts, which contained spells to help the dead reach the afterlife successfully, were only accessible to the elite. During the Middle Kingdom, Egyptians outside of the elite levels of society gained access to this funerary literature and began incorporating it into their own burials. Coffin Texts, as they are called by the scholars, expanded upon the Pyramid Texts, introducing new spells and incorporating slight changes to make them more relatable to the nobility. As seen in the image below, Coffin Texts could appear as paintings or inscriptions on the inside of the coffin. Due to the limited writing surfaces of some of these objects, the spells were often abbreviated, giving rise to long and short versions.
In contrast to the Pyramid Texts, which focus on the celestial realm, the Coffin Texts emphasize the subterranean elements of the afterlife ruled by Osiris in a place called the Duat . People of all classes had access to this afterlife, in which they would be judged by Osiris and his council according to their deeds in life. This realm is described as containing threatening beings, traps, and snares for which the deceased must be prepared. Spells in the Coffin Texts were intended to help the deceased contend with these impediments.
Stelae of the Middle Kingdom
The stelae of Ancient Egypt served many purposes, from funerary, to marking territory, to publishing decrees.
Describe the purposes of stelae in ancient Egyptian art
Key Points
- The earliest Egyptian stelae date back to the mid- to late third millennium BCE.
- Stelae are stone slabs that are inscribed, carved, or painted with imagery or text. While most were taller than they were wide, slab stelae took a horizontal dimension .
- Funerary stelae were generally built in honor of the deceased and decorated with their names and titles.
- Stelae also were used to publish laws and decrees, record a ruler’s exploits and honors, mark sacred territories or mortgaged properties, or to commemorate military victories.
- Obelisks , special forms of boundary stelae, were erected in pairs to mark the entrances of temples.
Key Terms
- obelisk :A tall, four-sided, narrow tapering monument which ends in a pyramid-like shape at the top.
- lintel :A structural horizontal block that spans the space or opening between two vertical supports.
- funerary :Of or relating to the interment or commemoration of the dead.
- stelae :Stone or wooden slabs, generally taller than they are wide, often erected for funerals commemorative purposes and decorated with images or text.
Egyptians were well known for their stelae, the earliest of which date back to the mid- to late third millennium BCE. Stelae are stone slabs that served many purposes, from funerary, to marking territory, to publishing decrees. Images and text were intimately interwoven and inscribed, carved in relief , or painted on the stelae. While most stelae were taller than they were wide, the slab stelae took a horizontal dimension and was used by a small list of ancient Egyptian dignitaries or their wives. The huge number of stelae surviving from ancient Egypt constitute one of the largest and most significant sources of information on those civilizations .
Funerary stelae were generally built in honor of the deceased and decorated with their names and titles. While some funerary stelae were in the form of slab stelae, this funerary stelae of a bowman named Semin (c. 2120-2051 BCE) appears to have been a traditional vertical stelae.
Slab stelae, when used for funerary purposes, were commonly commissioned by dignitaries and their wives. They also served as doorway lintels as early as the third millennium BCE, most famously decorating the home of Old Kingdom architect Hemon.
Stelae also were used to publish laws and decrees, to record a ruler’s exploits and honors, mark sacred territories or mortgaged properties, or to commemorate military victories. Much of what we know of the kingdoms and administrations of Egyptian kings are from the public and private stelae that recorded bureaucratic titles and other administrative information. One example of such stelae is the Annals of Amenemhat II , an important historical document for the reign of Amenemhat II (r. 1929–1895 BCE) and also for the history of Ancient Egypt and understanding kingship in general.
Many stelae were used as territorial markers to delineate land ownership. The most famous of these would be used at Amarna during the New Kingdom under Akhenaten. For much of Egyptian history, including the Middle Kingdom , obelisks erected in pairs were used to mark the entrances of temples. The earliest temple obelisk still in its original position is the red granite Obelisk of Senusret I (Twelfth Dynasty) at Al-Matariyyah in modern Heliopolis. The obelisk was the symbol and perceived place of existence of the sun god Ra.
Architecture of the Middle Kingdom
When Egypt had military and political security and vast agricultural and mineral wealth, its architecture flourished.
Describe the major building projects initiated during Egypt’s Middle Kingdom
Key Points
- The reign of Amenemhat III is known for its exploitation of resources and construction of the so-called “Black Pyramid .”
- Ancient Egyptian architects used sun-dried bricks, fine sandstone, limestone , and granite for their building purposes, though typically reserved stone for temples and tombs.
- Hieroglyphic and pictorial carvings in brilliant colors were abundantly used to decorate Egyptian structures.
- Senusret III is known for his construction of massive forts to defend the region after his many military campaigns.
- The Karnak Temple Complex is an example of fine architecture that was begun during the Middle Kingdom and continued through the Ptolemaic period.
Key Terms
- hieroglyph :An element of an Egyptian writing system.
As the pharaohs of the Middle Kingdom restored the country’s prosperity and stability, there was a resurgence of building projects. When Egypt had secured military and political security and vast agricultural and mineral wealth, its architecture flourished. Grand tombs in the form of pyramids continued to be built throughout the Middle Kingdom, along with villages, cities, and forts. The reign of Amenemhat III is especially known for its exploitation of resources, in which mining camps—previously only used by intermittent expeditions—were operated on a semi-permanent basis. A vast labor force of Canaanite settlers from the Near East aided in mining and building campaigns.
Ancient Egyptian architects used sun-dried bricks, fine sandstone, limestone, and granite for their building purposes. As in the Old Kingdom, stone was most often reserved for tombs and temples, while bricks were used for palaces, fortresses, everyday houses, and town walls. Mud was collected from the nearby Nile River, placed in molds, and left to dry and harden in the hot sun until they formed bricks for construction. Architects carefully planned all their work, fitting their stones and bricks precisely together. Hieroglyphic and pictorial carvings in brilliant colors were abundantly used to decorate Egyptian structures, and motifs such as the scarab, sacred beetle, solar disk, and vulture were common.
The “Black Pyramid” of Amenemhat III
The Black Pyramid, the first to house both the pharaoh and his queens, was built for Amenemhat III (r.
1860–1814 BCE). It is one of the five remaining pyramids of the original eleven pyramids at Dahshur in Egypt. Originally named
Amenemhet is Mighty
, the pyramid earned the name “Black Pyramid” for its dark, decaying appearance as a rubble mound. Typical for Middle Kingdom pyramids,
the Black Pyramid, although encased in limestone, is made of mud brick
and clay instead of stone. The ground-level structures consist of the
entrance opening into the courtyard and mortuary temple, surrounded by
walls. There are two sets of walls; between them, there are ten shaft
tombs, which are a type of burial structure formed from graves built
into natural rock. The capstone of the pyramid was covered with inscriptions and religious symbols.
Workers’ Villages and Forts
Workers’ villages were often built nearby to pyramid construction sites. Kahun (also known as El-Lahun), for example, is a village that was associated with the pyramid of Senusret II. The town was laid out in a regular plan, with mud-brick town walls on three sides. No evidence was found of a fourth wall, which may have collapsed and been washed away during the annual inundation. The town was rectangular in shape and was divided internally by a mud brick wall as large and strong as the exterior walls. This wall divided about one third of the area of the town, and in this smaller area the houses consisted of rows of back-to-back, side-by-side single room houses. The larger area, which was higher up the slope and thus benefited from whatever breeze was blowing, contained a much smaller number of large, multi-room villas, indicating perhaps a class separation between workers and overseers. A major feature of the town was the so-called “acropolis” building; its column bases suggest its importance.
Senusret III was a warrior-king who helped the Middle Kingdom reach its height of prosperity. In his sixth year, he re-dredged an Old Kingdom canal around the first cataract to facilitate travel to upper Nubia, using this to launch a series of brutal campaigns. After his victories, Senusret III built a series of massive forts throughout the country to establish the formal boundary between Egyptian conquests and unconquered Nubia. Buhen was the northernmost of a line of forts within signaling distance of one another. The fortress itself extended more than 150 meters along the west bank of the Nile, covering 13,000 square meters, and had within its wall a small town laid out in a grid system. At its peak, it probably had a population of around 3500 people. The fortress also included the administration for the whole fortified region. Its fortifications included a moat three meters deep, drawbridges, bastions, buttresses , ramparts, battlements, loopholes, and a catapult. The walls of the fort were about five meters thick and ten meters high .
The Karnak Temple Complex
The Karnak Temple Complex is an example of fine architecture that was begun during the Middle Kingdom and continued through the Ptolemaic period. Built by Senusret I, it was comprised of a vast mix of temples, chapels, pylons , and other buildings. The White Chapel, also referred to as the Jubilee Chapel, is one of the finest examples of architecture during this time. Its columns were intricately decorated with reliefs of a very high quality. Later in the New Kingdom, the Chapel was demolished; however, the dismantled pieces were discovered in the 1920s and carefully assembled into the building that is seen today.
- Curation and Revision. Provided by : Boundless.com. License : CC BY-SA: Attribution-ShareAlike
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2.6: Speaking to a Camera- Making Eye Contact is Key
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Zoom in, Zoom out.
But never lose your focus.
Shikha Td, poet
Let’s face it, the pandemic elevated the need for video meetings and presentations, and it is likely they are here to stay. Many businesses are doing interviews remotely and several have moved to online platforms for training. This chapter will talk about best practices for using a camera in a live online format and in a recorded and then playback format.
The first thing you need to do is fully understand the context. Will you be recording and uploading, or will you present live? Is this a one-time presentation? Will this presentation be recorded so others can see it online? All these factors will make a difference.
Make sure you have adequate lighting.
I once made a 30-minute video recording of a lecture to realize later that I didn’t have the equipment set up correctly. The sound was not recorded. I was so frustrated because I had to do the entire thing over again. Avoid my mistake and record a test segment and then make sure it works before you record too much of your speech. Look at the playback and consider whether the lighting is good enough for the audience to see your face.
If you have an important conference on Zoom or Teams, check your equipment beforehand. The first time I taught online, I practiced with my family. I practiced making the slides work, checking the microphone, the lighting, and the camera angle. Most accounts will allow you to sign up for a temporary free account if you need to practice outside of school or work.
Record yourself speaking for a minute and then play it back. Is the sound OK? Can the audience hear you clearly? If not, adjust your microphone. Sometimes the sound echoes in the room or there are too many background noises. Make sure your audio is clear and there is very little interference.
Some setups work better using a headset and microphone and some computers and cell phones work well with the main microphone. In many programs, you can go in through settings and adjust to the sound.
I had a student give his speech using his computer and he set his phone on the desk beside his laptop. He kept getting audio notifications beeping throughout his speech. It was very distracting.  If you are working from home, tell your roommates, siblings, and friends you will need the space to be quiet. Closing the window and closing the door can help keep out some ambient sound.
This may mean you set your laptop on a stack of books, or you readjust your web camera on your desktop. However it is accomplished, make sure the camera is level or a little higher than your eyes. The most common mistake that I see is when someone leans the laptop lid back making it look like the camera is looking up the nose of the speaker.
Eye contact is established when you look at the camera directly. If it helps, draw a smiley face and put it on your camera to remind you to look in the “eyes” of your audience.
Tape your note cards to the top of your computer screen or hang them on something behind your computer. Place your cards so you never have to look down to see your notes.
Depending on the type of speech, you want to frame your head or do a 3/4 shot. It is best to frame the shot so the audience can see your gestures.
Notice how Senator Elizabeth Warren uses the 3/4 shot so you can see her gestures. Notice how she looks directly into the camera.
President Barack Obama gave his speech with a headshot frame. Notice how his great “eye contact” make it look like he is talking to you directly.
You should read the situation on this one, but if in doubt stand. You will have better air support if you are standing. You will also gesture more freely.
You should have open gestures. If you do a tight headshot, you will not see your gestures, so you need to pull the camera back and gesture higher.
If you are having a friend record you with a handheld device, make sure that the zoom is kept in one place. Sometimes bored camera operators will zoom in and out making for a very unprofessional recording.
Your ending will leave a lasting impression so do not leave it to chance. Plan out the exact last words and then resist the temptation to tack on any last comments. Speech endings are always challenging, they are even more difficult online.
If you are recording your video in advance to be played back later. Edit the part where your arm reaches up to turn on and turn off your recording device. If appropriate, use the feature that allows you to put words on the screen if important words or concepts are relevant.
Here is a video I found helpful (mostly) about lighting setup and camera placement. I particularly like her advice to wear pants.
During the 2020 pandemic when many things went viral, Toastmasters held its speech contests virally. Notice how this creative speaker made the most of the situation to earn him the first-place win.
Remember This!
I want to hear from you.
Do you have an activity to include?
Did you notice a typo that I should correct?
Are you planning to use this as a resource and do you want me to know about it?
Do you want to tell me something that really helped you?
Carlucci, C. (2020). How to look better on video calls/facetime/Zoom. [Video] YouTube. https://www.youtube.com/watch?v=7ppTAA-1tm0 Standard YouTube License.
Carr, M. (2020). 2020 Toastmasters World Champion of Public Speaking: Mike Carr. [Video] YouTube. https://www.youtube.com/watch?v=TA7awoBxBxc Standard YouTube License.
Obama, B. (2020). President Barack Obama’s commencement speech to class of 2020 | Full Speech [Video] YouTube. https://www.youtube.com/watch?v=sTFWC1PiLVE Standard YouTube License.
National Speech and Debate Association. (2019). Senator Elizabeth Warren Celebrates National Speech and Debate Education Day! [Video] YouTube. https://www.youtube.com/watch?v=KYNO7cAzuYQ Standard YouTube License.


16: Markov Processes
A Markov process is a random process in which the future is independent of the past, given the present. Thus, Markov processes are the natural stochastic analogs of the deterministic processes described by differential and difference equations. They form one of the most important classes of random processes.
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- 16.1: Introduction to Markov Processes
- A Markov process is a random process indexed by time, and with the property that the future is independent of the past, given the present. Markov processes, named for Andrei Markov, are among the most important of all random processes. In a sense, they are the stochastic analogs of differential equations and recurrence relations, which are of course, among the most important deterministic processes.


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abstinence
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not engaging in sexual activity
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affect heuristic
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mental shortcut that causes us to use emotions more than other factors when making a decision
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affective empathy
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being able to vicariously experience someone’s emotions
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affective theory of mind
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ability to draw conclusions about another person’s feelings
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apprenticeship
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paid position involving a combination of on-the-job training, mentoring, and classroom learning
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availability heuristic
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mental shortcut that helps us think of memorable information quickly and use it to make decisions
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barrier method
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type of contraception and STI prevention that prevents exchange of bodily fluids
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cisgender
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describes an individual who has a gender identity compatible with their sex assigned at birth
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civil service job
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government job that doesn’t involve political appointments or legal work; examples include urban planners, budget analysts, sanitation workers, and firefighters
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cognitive empathy
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being able to adopt someone’s point of view
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cognitive theory of mind
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ability to draw conclusions about another person’s intentions and beliefs
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communal norm
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helping others with no expectation of reciprocity
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convergent thought
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using a known solution to deal with a problem
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dental dam
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thin piece of latex of polyurethane that’s placed over the vulva or anus prior to oral sex
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dialectical thinking
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ability to remain practical, flexible, and open to the multiple factors and solutions
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divergent thought
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creating new solutions to deal with a problem; related to creativity
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emerging adulthood
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beginning stages of adulthood characterized by the achievement of new legal statuses as well as intense exploration of career options, beliefs, and relationships
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epiphyses
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ends of long bones like the ones in arms and legs
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exchange norms
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either repaying a debt or anticipating future benefit when deciding whether to help another
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experiential learning
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education that involves “learning by doing” and reflecting on the learning experience
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familiarity heuristic
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mental shortcut that makes us feel more comfortable with things or people we have experience with than with those we have yet to experience or meet
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future orientation
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ability to think about and plan for what lies ahead, including predicting the consequences of one’s actions
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gender expression
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external manifestation of a person’s gender identity, including the way a person dresses, uses their voice, and presents themselves to the world
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heuristic
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mental shortcut used to process information more quickly
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informal education
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education that occurs without a set curriculum
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obesity
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when an individual’s BMI is over 30; correlated with many health issues including high blood pressure, diabetes, and heart disease
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obesity paradox
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observation that people with a high BMI who develop heart disease sometimes live longer and are in better health than people with a lower BMI who develop heart disease
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post-formal thought
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stage beyond formal operations that is more flexible, complex, and capable of recognizing more than one objectively correct answer to problem
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pre-exposure prophylaxis (PreP)
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medication taken regularly to reduce the chances of getting HIV
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presbyopia
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difficulty seeing things close up due to changes in the lens of the eye
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reflective thought
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logical thought processes involving continuous, active evaluation of information and beliefs using evidence and past experiences
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relativism
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idea that cultural standards, values, and ethics are a product of the time and cultural context within which they developed, and what is proper, relevant, or true for one individual may not be for another
-
scarcity heuristic
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mental shortcut that causes us to believe that the rarer or less available something is, the more valuable it is
-
sexual assault
-
nonconsensual sexual contact
-
sexual harassment
-
unwanted physical or verbal attention with a sexual basis
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sexually transmitted infection (STI)
-
disease caused by certain types of viruses, bacteria, or microorganisms and spread through contact with bodily fluids such as semen and blood
-
spermicide
-
substance that kills sperm cells
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substance abuse
-
use of drugs or alcohol in excess or for purposes for which they weren’t intended, leading to problems in physical, social, and/or psychological functioning, often termed substance misuse or problematic substance use in order to avoid stigma of "abuse"; often related to substance use disorder (SUD)


1.4: Defining the Boundaries and Symbols
As we will see later in this text, California can be viewed several ways; political, economic, cultural, geographically, and even linguistically. As mentioned earlier, the earlier boundaries of California were defined by Spanish claims of Mexico as part of the province of Alta California. The northern boundary of Spanish claims was set at 42°N latitude by the Adams–Onis Treaty of 1819. The states of Nevada and Utah, also originally part of Alta California, also use that line for their northern boundaries. The southern boundary, between California and Mexico, was established by the Treaty of Guadalupe Hidalgo that ended the Mexican-American War in 1848. The line is about 30 miles north of the former Alta California southern boundary. The eastern boundary consists of two straight lines: a north–south line from the northern border to the middle of Lake Tahoe, and a second line angling southeast to the Colorado River. From that point, 14 miles (23 km) south-southwest of Davis Dam on Lake Mohave, the southeast boundary follows the Colorado River to the international border west of Yuma. The eastern and south-eastern boundaries were decided upon during the debates of the California Constitutional Convention in 1849.
The following image, figure 1.7, identifies some of the landform divisions that have helped define not just the physical boundaries, but the greater regions within our state. In future units, the text will define each region of the state, including the physical, geological, cultural and in many instances even specific case studies of towns and people within that region.
The Symbols of California
The Great Seal of the State of California was adopted at the California state Constitutional Convention of 1849, (figure 1.8), and has undergone design changes since then, the last being the standardization of the seal in 1937, in which the current seal still resembles (figure 1.9). Most recent seal shows Athena, (original named Minerva) in Greek mythology, the Goddess of wisdom and war, as she was born an adult. A California grizzly bear, the official state animal, feeding on grape vines, representing California's wine production; a sheaf of grain, representing agriculture; a miner, representing the California Gold Rush and the mining industry; and sailing ships, representing the state's economic power. The word Eureka (εύρηκα in Greek), meaning "I have found it", is the California state motto.
So how did we get to the current seal? It all started in 1928, due to the number of incorrect details that had appeared in the seal over the years, state printer Carroll H. Smith was authorized to prepare a new and correct seal. This seal was drawn by Los Angeles heraldic artist Marc J. Rowe who, among other corrections, narrowed the growing break in the mountains so that it appeared to be the Sacramento River, fringed by snow-capped Sierra, and not an arm of San Francisco Bay, as the 1876 seal made it appear. Nine years later, the 1937 standardized seal once again featured a widened gap of Golden Gate proportions, although it did keep Rowe's snow-capped Sierra Nevada that had replaced the barren foothills of previous editions of the seal. Both features have stayed in the official seal.
The 1937 standardization came about when state employees, wishing to print the seal on blotters for the State Fair, could not find any official design of the seal. This prompted a new law which established for the first time a definite pictured design with which the master die was 'substantially' to conform, and at the same time established the legality of all previous seals which were essentially the same as this one.
The symbols found within the Seal are often easily recognized, such as the State motto. Other symbols become just common knowledge such as the nickname, the Golden State. Other lesser-known symbols of California range from the state amphibian to the state vegetable. This text will continue to celebrate the diversity of California; however, it seems appropriate to list some the more obscure and trivial designated symbols of California.
|
Symbol |
California State |
|---|---|
|
Amphibian |
California Red-Legged Frog |
|
Animal |
California Grizzly Bear |
|
Bird |
California Quail |
|
Colors |
Blue & Gold |
|
Dance |
West-Coast Swing |
|
Dinosaur |
Augustynollphus |
|
Fabric |
Denim |
|
Fish |
Golden Trout |
|
Flower |
California Poppy |
|
Fossil |
Smilodon Californicus |
|
Fruit |
Avocado |
|
Gemstone |
Benitoite |
|
Gold Rush Town |
Bodie |
|
Mineral |
Gold |
|
Outdoor Play |
The Ramona Pageant |
|
Reptile |
Desert Tortoise |
|
Rock |
Serpentine |
|
Silver Ghost Town |
Calico |
|
Soil |
San Joaquin |
|
Sport |
Surfing |
|
Theater |
Pasadena Playhouse |
|
Trees |
Costal Redwood & The Giant Sequoia |
|
Vegetable |
Artichoke |
Figure 1.10 List of California State Symbols. [4]
[1] California State University, Northridge. (2011). California: Physical Features. csun.edu. Retrieved 6/15/2023 from: http://www.csun.edu/~cfe/maps/?C=N;O=A
[2] Illustration by Henry Mitchell is in the public domain.
[3] Fichier d’origine is in the public domain
[4] Table by Jeremy Patrich