Each day, two out of the three teams in a class are randomly selected to participate in a MATHCOUNTS trial competition. What is the probability that Team A is selected on at least two of the next three days? Express your answer as a common fraction.
We can compute this a few ways, but the numbers seem small enough that we can go ahead and just compute the probability of A being selected all three days, and the probability of A being selected exactly 2 of the three days. Team A is selected on any given day with probability $\frac{2}{3}$, because there are ${3 \choose 2} = 3$ possible pairs of teams, and 2 of them contain A. So, there is a $\left(\frac{2}{3}\right)^3 = \frac{8}{27}$ chance of being selected all three days. Of being selected exactly twice, there is a $\frac{2}{3} \cdot \frac{2}{3} \cdot \frac{1}{3} \cdot {3 \choose 2} = \frac{4}{9}$ chance. Adding these two yields $\frac{8}{27} + \frac{4}{9} = \frac{8+12}{27} = \boxed{\frac{20}{27}}$.