Bayes rule Help me out please Show all work in order to rece

Bayes rule, Help me out please

Show all work in order to receive credit. This is a take home quiz that should be submitted at the start of class on Friday 9/25/2015. In our M&M; jar are 745 peanut M&M;\'s, 748 pretzel M&M;\'s and 979 peanut butter M&M;\'s. The published proportion of color vs M&M; types are shown below: Blue Orange Green Yellow Red Brown Peanut 0.23 0.23 0.15 0.15 0.12 0.12 Pretzel 0.167 0.167 0.167 0.167 0.167 0.167 Peanut Butter 0.2 0.2 0.2 0.2 0.1 0.1

Solution

Let

A = its is Peanut
B = it is Pretzel
C = it is Peanut butter

There are 745+748+979 = 2472 MM\'s all in all.

Thus,

P(A) = 745/2472 = 0.301375405
P(B) = 748/2472 = 0.302588997
P(C) = 979/2472 = 0.396035599

Now, thus,

P(B|Y) = P(B and Y) / P(Y)

As

P(B and Y) = P(B) P(Y|B) = 0.302588997*0.2 = 0.050532362

P(Y) = P(A) P(Y|A) + P(B) P(Y|B) + P(C) P(Y|C) = P(A) *0.15 + P(B)*0.167 + P(C)*0.2 = 0.174945793

Thus,

P(B|Y) = 0.288845828 [ANSWER]