What's the probability that the boy loves pizza?

Question

Problem

A survey of boy-girl couples suggests that in 20% of couples both the boy and the girl love pizza, in 5% of couples the girl loves pizza, but the boy doesn't, in 10% couples the boy loves pizza, but the girl doesn't.

If a boy is chosen randomly and he loves pizza, what's the probability that the girl loves pizza too?

Solution attempt

$A$ - the boy loves pizza, $B$ - the girl loves pizza

Then we know that $P(AB)=0.2$, we need $P(B|A)$, but I guess we don't know $P(A)$, we only know $P(\overline{A}B)$ and $P(A\overline{B})$, but I don't know what to do with that information.

How to solve that problem?

Answer 1

Note $P(A)=P(AB)+P(A\bar{B})=0.2+0.1=0.3$. You can do the rest.

Answer 2

Hint:

In $100$ couples:

• How many couples do you expect to see where both the boy and the girl love pizza?
• How many couples do you expect to see where the boy loves pizza but the girl does not?
• How many couples do you expect to see where the boy loves pizza?
• In what proportion of the couples where the boy loves pizza do you expect to see the girl loves pizza too?

More formally, the answers to these questions are $100 \,\mathbb P(AB)$, $100\, \mathbb P(A\overline B)$, $100 \,\mathbb P(A)$ and $\mathbb P(B \mid A)= \frac{100 \mathbb P(AB)}{100 \mathbb P(A)} = \frac{\mathbb P(AB)}{ \mathbb P(AB)+ \mathbb P(A\overline B)}$, all of which you can calculate