1. [5] David, Delong, and Justin each showed up to a problem writing session at a random time during the session. If David arrived before Delong, what is the probability that he also arrived before Justin?
Answer: $\sqrt{\frac{2}{3}}$ Let $t_{1}$ be the time that David arrives, let $t_{2}$ be the time that Delong arrives, and let $t_{3}$ be the time that Justin arrives. We can assume that all times are pairwise distinct because the probability of any two being equal is zero. Because the times were originally random and independent before we were given any information, then all orders $t_{1}<t_{2}<t_{3}, t_{1}<t_{3}<t_{2}, t_{3}<t_{1}<t_{2}, t_{2}<t_{1}<t_{3}$, $t_{2}<t_{3}<t_{1}, t_{3}<t_{2}<t_{1}$ must all be equally likely. Since we are given that $t_{1}<t_{2}$, then we only have the first three cases to consider, and $t_{1}<t_{3}$ in two cases of these three. Thus, the desired probability is $\frac{2}{3}$.