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Question

At State University, 30% of the students are majoring in humanities, 50% in history and culture, and 20% in science. Moreover, according to figures released by the registrar, the percentages of women majoring in humanities, history and culture, and science are 75%, 45%, and 30%, respectively. Suppose Justin meets Anna at a fraternity party. What is the probability that Anna is a history and culture major?

Solution

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Let

A1A_1= Humanities

A2A_2= History and culture

A3A_3= Science

BB=Woman

Given:

P(A1)=30%=0.3P(A_1)=30\%=0.3

P(A2)=50%=0.5P(A_2)=50\%=0.5

P(A3)=20%=0.2P(A_3)=20\%=0.2

P(BA1)=75%=0.75P(B|A_1)=75\%=0.75

P(BA2)=45%=0.45P(B|A_2)=45\%=0.45

P(BA3)=30%=0.3P(B|A_3)=30\%=0.3

Bayes's theorem:

P(AiB)=P(BAi)P(Ai)j=1kP(BAj)P(Aj)P(A_i|B)=\dfrac{P(B|A_i)P(A_i)}{\sum_{j=1}^k P(B|A_j)P(A_j)}

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