Three bottles of different sizes contain different compositions of red and blue

Question

Three bottles of different sizes contain different compositions of red and blue candy. The largest bottle contains eight red and two blue pieces, the mid-size bottle has five red and seven blue, the small bottle holds four red and two blue. A monkey will pick one of these three bottles, and then pick one piece of candy from it. Because of the size differences, there is a probability of 0.5 that the large bottle will be picked, and a probability of 0.4 that the mid-size bottle is chosen. Once a bottle is picked, it is equally likely that the monkey will select any of the candy inside, regardless of color. What is the probability that a blue candy is picked? If a blue candy is picked, what is the probability that the large bottle was selected?

Explanation / Answer

a)

Let

L = large bottle
M = midsize bottle
S = small bottle
B = blue candy
R = red candy

a)

By Bayes' Rule,

P(B) = P(L) P(B|L) + P(M) P(B|M) + P(S) P(B|S)

= 0.5*(2/10) + 0.4*(7/12) + 0.1*(2/6)

P(B) = 0.366666667 [answer]

****************

b)

P(L|B) = P(L) P(B|L) / P(B) = 0.5*(2/10) / 0.36666667 = 0.27272727 [answer]

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