In a recent election there were 1000 eligible voters. They were asked to vote on two issues, $A$ and B. The results were as follows: 250 people voted for A, 450 people voted for B, and 25 people voted for A and B. If one person is chosen at random from the 1000 eligible voters, find the following probabilities:
- The person voted for $A$ given that he voted for $B$.
- The person voted for $B$ given that he voted for $A$.
My attempt at solving this goes like this:
$P(A|B) = \frac{P(A \cap B)}{P(B)}$
$P(A) = \frac{250}{1000} = \frac{1}{4}$
$P(B) = \frac{450}{1000} = \frac{9}{20}$
$P(A \cap B) = \frac{25}{1000} = \frac{1}{40}$
Thus the probability of A given B should be: $P(A|B) = \frac{P(A \cap B)}{P(B)} = \frac{\frac{1}{40}}{\frac{9}{20}} = \frac{1}{18}$.
Which is apparently not the correct answer.
You forgot that the probability someone voted for $B$ is not just that they only voted $B$, but rather that the voted $B$ at all. This is $$ P(B) = \frac{450+25}{1000} = 0.475. $$