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# 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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Step 1
1 of 3

Let

$A_1$= Humanities

$A_2$= History and culture

$A_3$= Science

$B$=Woman

Given:

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

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

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

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

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

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

Bayes's theorem:

$P(A_i|B)=\dfrac{P(B|A_i)P(A_i)}{\sum_{j=1}^k P(B|A_j)P(A_j)}$

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