Bayes\'s theorem on conditional probabilities states that, if P(B) is not 0, the

Question

Bayes's theorem on conditional probabilities states that, if P(B) is not 0, then P(A given B) = P(B given A) × P(A) P(B) . Yesterday car lot Alpha had two Toyotas and one Chevrolet for sale. Car lot Beta had three Toyotas and five Chevrolets for sale. This morning Alan bought a car, choosing one of the two lots at random and then choosing a car at random from that lot. Leave your answers as fractions.

(a) What is the probability that Alan bought from Alpha? Incorrect: Your answer is incorrect.

(b) If Alan bought from Alpha, what is the probability he bought a Toyota? Incorrect: Your answer is incorrect.

(c) If Alan bought from Beta, what is the probability he bought a Toyota? Incorrect: Your answer is incorrect.

(d) It can be shown that the probability that Alan bought a Toyota is 25/48. What is the probability Alan bought from Alpha given that he bought a Toyota? Note: In this situation, Bayes's theorem says P(Alpha given Toyota) = P(Toyota given Alpha) × P(Alpha) P(Toyota) .

Explanation / Answer

a. P(Alpha) = 1/2

b. P(Toyota | Alpha) = 2/3

c. P(Toyota | Beta) = 3/8

d. P(Toyota) = P(Alpha) * P(Toyota | Alpha) + P(Beta) * P(Toyota | Beta) = (1/2 * 2/3) + (1/2 * 3/8)

= 25/48

Using Bayes' theorem,

P(Alpha | Toyota) = P(Alpha) * P(Toyota | Alpha) / P(Toyota) = (1/2 * 2/3) / (25/48) = 16/25

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