At a certain university, 20% of students only major in literature, 15% of studen

Question

At a certain university, 20% of students only major in literature, 15% of students only major in history, and 8% of students major in both literature and history. The literature-only majors spend a semester abroad with probability 0.32, the history-only majors spend a semester abroad with probability 0.49, and the literature-history double-majors study abroad with probability 0.93. Students of other major's study abroad with probability 0.04. If a student from this university spends a semester studying abroad, what is the probability he or she is a double-major in literature and history?

Explanation / Answer

Let O shows the event that student has other major. Let L shows the event that student major in literature, H shows the event that student major in history so we have

P(only L) =0.20, P(only H) = 0.15, P(L and H) = 0.08, P(O) = 1 - 0.20 - 0.15 - 0.08 = 0.57

Let A shows the event that student stdy abroad. So we have

P(A|only L) = 0.32

P(A|only H) = 0.49

P(A| L and H) = 0.93

P(A|O) = 0.04

By the total probability, the probabiltiy that student study in abroad is

P(A) =P(A|only L) P(only L)+P(A|only H) P(only H)+ P(A| L and H)P(L and H) + P(A|O)P(O)=0.32*0.20 + 0.49*.0.15 + 0.08 * 0.93 + 0.57*0.04 = 0.2347

By the Baye's theorem the conditional probability is

P((L and H)|A) =[ P(A| L and H)P(L and H)] / P(A) =[0.93* 0.08] / 0.2347 =0.3170

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