Answer to Question #123021 in Statistics and Probability for Rohit kumar prasad

Question #123021
Student population of a university has 30% asian, 40% american, 20% european and 10% african students. It is known that 40% of all asian students, 50% of all american students, 60% of all european students and 20% of all african students are girls. Find the probability that a girl chosen at random from the university is an asian?
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Expert's answer
2020-06-21T16:49:28-0400

Let A1A_1 denote asian student, A2A_2 denote american student, A3A_3 denote european student, A4A_4 denote african student, and GG denote that a tudent is a girl.

Given

P(A1)=0.3,P(A_1)=0.3,

P(A2)=0.4,P(A_2)=0.4,

P(A3)=0.2,P(A_3)=0.2,

P(A4)=0.1,P(A_4)=0.1,

P(GA1)=0.4,P(G|A_1)=0.4,

P(GA2)=0.5,P(G|A_2)=0.5,

P(GA3)=0.6,P(G|A_3)=0.6,

P(GA4)=0.2P(G|A_4)=0.2


P(G)=P(A1)P(GA1)+P(A2)P(GA2)+P(G)=P(A_1)P(G|A_1)+P(A_2)P(G|A_2)+

+P(A3)P(GA3)+P(A4)P(GA4)=+P(A_3)P(G|A_3)+P(A_4)P(G|A_4)=

=0.3(0.4)+0.4(0.5)+0.2(0.6)+0.1(0.2)=0.46=0.3(0.4)+0.4(0.5)+0.2(0.6)+0.1(0.2)=0.46

Use the Bayes' Theorem


P(A1G)=P(A1)P(GA1)P(G)=P(A_1|G)=\dfrac{P(A_1)P(G|A_1)}{P(G)}==0.3(0.4)0.46=6230.260870=\dfrac{0.3(0.4)}{0.46}={6\over 23}\approx0.260870


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