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

Question

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? Answer: 0.75

1. 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?

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

(Please answer questions 1 and 2)

Explanation / Answer

Jar A has 75% red and 25% blue marbles.

Jar B has 25% red and 75% blue marbles.

Let R and B be the events denoting selection of red and blue marble respectively.

Then P(Jar A|R) = P(R|Jar A) P(Jar A) / [P(R|Jar A) P(Jar A) + P(R|Jar B) P(Jar B)]

= 0.75*0.5 / [0.75*0.5 + 0.25*0.5] = 0.75 .

1. Player 1 predicted Jar A. I am the second player in this sequential guessing task and I predicted red.

Then P(Jar A|Red marble)

= P(Red|Jar A)P(Jar A) / [P(Red|Jar A)P(Jar A) + P(Red|Jar B)P(Jar B)]

= 0.75*0.5 / [0.75*0.5 + 0.25*0.5] = 0.75 .

2. P(Red|Jar A) = 0.75, P(Blue|Jar A) = 0.25.

Since odds of prediction of blue marble given jar A to selection of red marble is 0.25/0.75 = 1/3, so my prediction is not rational.

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