Question

Suppose there are two full bowls of cookies. Bowl #1 has 20 chocolate chip and 30 oatmeal, while bowl #2 has 30 chocolate chip cookies and 20 oatmeal cookies. Our friend Bob picks a bowl at random, and then picks a cookie at random. We may assume there is no reason to believe Bob treats one bowl differently from another, likewise for the cookies. The cookie turns out to be a chocolate chip one. How probable is it that Bob picked it out of Bowl #2?

Answer 1

Let

A = the event of chocolate chip

Let B₁ and B₂ be the events of selecting Bowl #1 and Bowl #2 respectively.

Since selecting Bowl # 1 and Bowl # 2 are equally likely:
P(B₁) = 0.5

P(B₂) = 0.5

Bowl # 1:

chocolate chip = 20

oatmeal = 30

Total cookies = 50

So,

P(A/B₁) = 20/50 = 0.4

Bowl # 2:

chocolate chip = 30

oatmeal = 20

Total cookies = 50

So,

P(A/B₂) = 30/50 = 0.6

By Bayes Theorem:

So,

Answer is:

0.6