In a drawer Sandy has $5$ pairs of socks, each pair a different color. On Monday Sandy selects two individual socks at random from the $10$ socks in the drawer. On Tuesday Sandy selects $2$ of the remaining $8$ socks at random and on Wednesday two of the remaining $6$ socks at random. Find the probability that Wednesday is the first day Sandy selects matching socks.
We start by choosing the third-day pick. There is a $1\times\frac{1}{9}=\frac{1}{9}$ chance that a matching pair is picked.
Next, on the first day, there is a $1\times\frac{6}{7}=\frac{6}{7}$ chance that a mismatched pair is picked.
Then, the second day, there are totally $C_6^2$ ways to pick up two socks and two ways will result a match. Hence, the probability of picking a pair of unmatched socks is $\left(1- 2 / C_6^2\right)=\frac{13}{15}$
Therefore, the answer is $$\frac{1}{9}\times\frac{6}{7}\times\frac{13}{15}=\frac{26}{315}$$