I have a spinner that lands on 1 with a probability of $\frac{1}{10}$, 2 with a probability of $\frac{2}{10}$, 3 with a probability of $\frac{3}{10}$, and 4 with a probability of $\frac{4}{10}$. If Phil and Sarah both spin the spinner, what is the probability that they get the same number?
This problem requires a little bit of casework.  There are four ways in which they can both get the same number: if they both get 1's, both get 2's, both get 3's or both get 4's. The probability of getting a 1 is $\dfrac{1}{10}$, so the probability that they will both spin a 1 is $\left(\dfrac{1}{10}\right)^2=\dfrac{1}{100}$. Similarly, the probability of getting a 2 is $\dfrac{2}{10}$, so the probability that they will both spin a 2 is $\left(\dfrac{2}{10}\right)^2=\dfrac{4}{100}$, the probability of getting a 3 is $\dfrac{3}{10}$, so the probability that they will both get a 3 is $\left(\dfrac{3}{10}\right)^2=\dfrac{9}{100}$ and the probability of getting a 4 is $\dfrac{4}{10}$, so the probability that they will both get a 4 is $\left(\dfrac{4}{10}\right)^2=\dfrac{16}{100}$. So our answer is $\dfrac{1}{100}+\dfrac{4}{100}+\dfrac{9}{100}+\dfrac{16}{100}=\frac{30}{100}=\boxed{\dfrac{3}{10}}$.