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Problem 3. (5-+5-+5-+10-25 points) Suppose there are two full box of marbles. The green box has 10 black marbles and 40 white marbles, while red box has 25 of each. I pick a box at random, and then pick a marble at random 1. Complete the following tree diagram (filling up the nodes on the 2 layers as well as the probabilities on all of the edges) a marble chosen at random 2. We have two hypothesis: Ho: The marble pikced from green box;H1: The marble picked from red box. Whats your belief for each hypothesis at this moment (whats the probability for each hypothesis to happe)? 3. Given the marble from green box, what is the likelihood that the marble is black? 4. Now the marble turns out to be black. What is the updated belief for the two hypotheses (probabilities for each hypothesis to happen) now? (Hint: Solve the problem by using the Bayes rule.)

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Answer #1

Black Box 5White Marhle 8ox White Marble Hi: Marble em Mom Red Box Oak nx) Blask oxBlock marole) 2+5 P (Rod. Box | Black Mabu ) = 5

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