Question

In a small town, there are three bakeries. Each of the bakeries bakes twelve cakes

per day. Bakery 1 has two dierent types of cake, bakery 2 three dierent types,

and bakery 3 four dierent types. Every bakery bakes equal amounts of cakes of

each type.

You randomly walk into one of the bakeries, and then randomly buy two cakes.

(a) What is the probability that you will buy two cakes of the same type?

(b) Suppose you have bought two dierent types of cake. Given this, what is the

probability that you went to bakery 2?

Answer 1

Bakery 1 - 6 cakes each of the 2 types

P(selecting 2 cakes of same types) = Number of types x Number of ways in which 2 cakes of same type can be chosen / Number of ways to select 2 cakes from 12

= 2 x 6C2 / 12C2

= 2 x 15 / 66

= 5/11

Bakery 2 - 4 cakes each of the 3 types

P(selecting 2 cakes of same type from bakery 2) = 3x4C2/12C2

= 3 x 6 / 66

= 3/11

Bakery 3 - 3 cakes each of the 2 types

P(selecting 2 cakes of same type from bakery 3) = 4x3C2/12C2

= 4 x 3 / 66

= 2/11

a) P(selecting 2 cakes of same type) = (1/3 x 5/11) + (1/3 x 3/11) + (1/3 x 2/11)

= 10/33

= 0.3030

b) Bayes' Theorem: P(A | B) = P(A & B)/P(B)

P(bakery 2 | two cakes of different types) = P(bakery 2 and 2 cakes of different types) / P(two cakes of different types)

= [1/3(1 - 3/11)] / (1 - 10/33)

= [1/3x(8/11)]/(23/33)

= 8/23

= 0.3478