1. You are a participant in a sequential guessing task. You draw marbles from an

Question

1. You are a participant in a sequential guessing task. You draw marbles from an opaque jar with red and blue marbles in it. You know that the jar is one of two jars, with equal probability. Jar A has 75% red marbles. Jar B has 75% blue marbles. Using Bayes Theorem and assuming you are the first to predict, what is the probability of Cup A given the fact that you selected a red marble?

2. Assume you are the second player to predict, and player 1 predicted Jar A. Using Bayes’ Theorem, what is the probability of Jar A given the fact that you selected a red marble?

3. If you pick third and draw a blue marble, but the first two players predicted Cup A, what is your rational prediction? Why?

Explanation / Answer

1:

Let A shows the event that jar A is selected and B shows the event that bar B is selected. So

P(A) = P(B) = 0.5

Let R shows the event that a red marble is selected and Bl shows the event that a blue marble is selected. So we have

P(R|A) = 0.75, P(Bl | B) = 0.75

Using complement rule,

P(Bl |A) = 1 - P(R|A) = 0.25, P(R|B ) = 1 - P(Bl |B) = 1 -0.75 = 0.25

By tle law of total probability, we have

P(R) = P(R|B)P(B) + P(R |A)P(A) = 0.75 *0.5 + 0.25 * 0.5 = 0.5

By the Baye's theorem,  the probability of Cup A given the fact that you selected a red marble is

P(A |R ) = [ P(R |A)P(A) ] / P(R) = [ 0.75 *0.5 ]/ 0.5 = 0.75

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