Another socks in drawers problem

Question

I'm stuck with this question (sorry if it's ridiculously simple, I'm new to this): A person keeps his socks in two drawers. The first drawer contains 8 socks of which 4 are white. The second drawer contains 10 socks of which 8 are white. He chooses one drawer randomly and picks two socks from that drawer. Find the probability that both socks are white. I've calculated the probability for getting 2 white socks from drawer 1 (3/14) or from drawer 2 (14/45) (using the formula for B, given that A has happened). But what now??

Answer 1

Let $D_1$ be that our person picked the first drawer. Let $D_2$ be that our person picked the second drawer. Let $W_1$ be that the first sock selected is white. Let $W_2$ be that the second sock selected is white.

Note that $D_1\cap D_2=\emptyset$ and that $D_1\cup D_2=\Omega$

We are tasked with calculating $Pr(W_1\cap W_2)$, that both socks selected are white.

By total probability and by multiplication principle of probability we have:

$Pr(W_1\cap W_2)=Pr(D_1)Pr(W_1\mid D_1)Pr(W_2\mid D_1\cap W_1) + Pr(D_2)Pr(W_1\mid D_2)Pr(W_2\mid D_2\cap W_1)$

$=\frac{1}{2}\cdot \frac{4}{8}\cdot \frac{3}{7} + \frac{1}{2}\cdot\frac{8}{10}\cdot\frac{7}{9}$

This may all be visualized through the use of a tree diagram.