In a deck of $n$ cards, 30% have white on both sides, 50% have black and white sides, and 20% of cards have black on both sides. The deck is then shuffled fairly and a random card is drawn and set on the table. If the card is black, then what is the probability that the other side is white? (Pitman's Probability, 1.4.8)
Here's what I thought:
We have that P(WW) = 0.3, P(BW) = 0.5, and P(BB) = 0.2. We want to find P(W | B).
P(W | B) = $\frac{P(BW)}{P(B)}$ = $\frac{0.5}{0.5 + 0.2}$ = $\frac{5}{7}$.
However, the solution manual says the answer is $\frac{0.5}{0.5 + 2*0.2}$. According to this, we need to consider P(BB) two times, rather than just once. Why is this? Isn't the probability of a card having black on one side the union of Black & White cards and Black & Black cards? What am I missing here? Thank you.