# Question #7d4be

Aug 8, 2016

$.5795$

#### Explanation:

Let $B$ be the event of raining and $A$ be the event that the theater is full

we are asked for $p \left(A\right)$, the probability of the theater being full.
we are given the following
$p \left(A | B\right) = .8$ The probability of being full given it rains
$p \left(A | \neg B\right) = .45$ The probability of being full given it does not rain aka sunny day
$p \left(B\right) = .37$ The probability of raining

Based on the law of total probability we know that
$p \left(A\right) = p \left(A , B\right) + p \left(A , \neg B\right) =$
$p \left(A | B\right) p \left(B\right) + p \left(A | \neg B\right) p \left(\neg B\right)$

now we just plug in
$p \left(A\right) = .8 \cdot .37 + .45 \cdot .63 = .5795$