Dr. XY thinks he can guess your sex by only knowing your height. His rule is sim

Question

Dr. XY thinks he can guess your sex by only knowing your height. His rule is simple: if your height is greater than 172 cm he predicts you are male. If your height is less than 172 he predicts you are female. You look up the statistics for male and female heights around the world for 20 year olds (we are going to assume that height doesn’t change much after this age) and find that about 75% of males are greater than 172 cm and 20% of females are greater than 172 cm. The worldwide sex ratio of males to females is 101:100). What is the probability that a randomly chosen individual who is over 172 cm is a male? What is the probability that an individual who is under 172 cm is a female?

Explanation / Answer

Let H shows the event that height is greater than 172 cm and L shows the event that height is less than 172 cm. Let M shows the event that person is male anf F shows the event that person is female. So we have following probabilites;

P(M) = 101 /(101+100) =101/201, P(F) = 100 /(101+100) =100/201

and following conditional probabilites:

P(H |M) = 0.75 , P(H|F) = 0.20

P(L|M) = 1- P(H|M) = 0.25, P(L|F) =1- P(H|F) = 0.80

By the law of total probability we have

P(H)= P(H|M)P(M) + P(H|F)P(F) = 0.75*(101/201) + 0.20*(100/201) = 0.4764

P(L)= P(L|M)P(M) + P(L|F)P(F) = 0.25*(101/201) + 0.80*(100/201) = 0.5236

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What is the probability that a randomly chosen individual who is over 172 cm is a male?

Here we need to find the probability P(M|H). By the Bayes thoerem we have

P(M|H) = [P(H|M) P(M) ] / P(H) = 0.7911

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What is the probability that an individual who is under 172 cm is a female?

Here we need to find the probability P(F|L). By the Bayes thoerem we have

P(F|L) = [P(L|F) P(F) ] / P(L) = 0.7601

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