Rutgers Business School, Introduction to Probability, 26:960:575. Homework 1. Prepared by Daniel Pirutinsky. Feburary 1st, 2016

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1 Rutgers usiness School, Introduction to Probability, 6:960:575 Homework Prepared by Daniel Pirutinsky Feburary st, 06 Introduction Suppose we have two boxes, labeled and. Each box contains some amount of white balls and some amount of black balls. Let i denote the proportion of white white balls in box i and i the proportion of black balls in box i. For example if ox has 0 white balls and 40 black balls then =. and =.8 A dealer will choose a box (uniformly at random) and our job is to guess which box the dealer chose. If we guess correctly we will win a reward of R dollars. If we are wrong, we will lose and have to pay L dollars. e are also given the option to sample (with replacement) a ball from the box the dealer chose to help inform us which box it really is. e can sample as many time as we want, but we must pay C dollars each time. The question is, what is the strategy that maximizes our expected reward? efore we analyze this question, we will first define some more terms. Let D denote the box that the dealer chose. In our case, either D = if the dealer chose box, or D = if the dealer chose box. Let D i will be the initial probability that the dealer chose box i, which we take to be D = D =.5 Let O denote the ordered samples that we have already observed. For example, in the beginning O =, if we sampled three times and saw a white, black, and then a white, O = Let S denote the next observation. In our case we may sample and either S = if we observe a white ball, or S = if we observe a black ball. Summary of variables Proportion of white balls in box Proportion of black balls in box Proportion of white balls in box Proportion of black balls in box R = 00 Reward if we guess correctly L = 50 Loss if we guess incorrectly C = 0 Cost per sample D =.5 Initial probability (prior) that the dealer chose box D =.5 Initial probability (prior) that the dealer chose box Summary of random variables D {, } ox that the dealer chose O {,,,,,,,,...} Ordered observations S {, } Next observation

2 e can visuallize our game as follows, Problem 0 lack 0 hite 0 lack 0 hite ox ox Find the strategy for the case where box has 0 black balls and 0 white balls, and box has 0 black balls and no white balls.

3 e have: = =.5 and = 0, =. e will find the optimal policy by examining each action in turn. First we will calculate the expected reward if we guess without sampling at all. If we guess box, the with probability D we will gain R = 00 and with probability D we will lose L = 50, so the expected reward can be written as, E[reward guess O = ] = R P (D = ) L P (D = ) = R D L D If guess will be the same. Indeed: = R L = = 5 And so the expected reward can be written as, E[reward guess O = ] = L P (D = ) + RL P (D = ) = L D + R D e update the picture, = L + R = = 5 3

4 +5 (or ) If we choose to sample, we will pay C = 0, and with probability P (S = ) we will observe a white, and with probability P (S = ) we will observe a black. To find these probabilities we can use the law of total probability and we have, P (S = ) = P (S = D = ) P (D = ) + P (S = D = ) P (D = ) = D + D = + 0 = 4..[...and P (S = ) = P (S = ) = ]. -.. P (S = ) = P (S = D = ) P (D = ) + P (S = D = ) P (D = ) = D + D = + = 3 4 If we observed a white ball, we need to determine the probablity that the dealer chose box : P (D = S = ), and the probability that the dealer chose box : P (D = S = ). Using ayes Theorem (and our previous calculations) we have, P (D = S = ) = and P (S = D = ) P (D = ) P (S = ) P (D = S = ) = P (D = S = ) = 0 = 4 = 4

5 So if we observe a white ball and we decide to guess, we will guess box, and with probability P (D = S = ) = we will win R = 00 but have paid C = 0 to sample, so we will net 80. Since we are sure that the dealer chose box, we will not sample again. If we observed a black ball, we need to determine the probablity that the dealer chose box P (D = S = ), and the probability that the dealer chose box, P (D = S = ). Using ayes Theorem (and our previous calculations) we have, P (D = S = ) = and P (S = D = ) P (D = ) P (S = ) P (D = S = ) = P (D = S = ) = 3 = 3 4 = 3 So if we observe a black ball and we decide to guess, we will guess box, and our expected reward is, E[reward guess S = ] = R P (D = S = ) L P (D = S = ) 0 = = = 50 0 = (or ) Following the path down from, if we choose to sample again, we will pay another C = 0, and with probability P (S = O = ) we will observe a white, and with probability P (S = O = ) we will observe a black. P (S = O = ) = P (S = (O = D = )) P (O = D = ) + P (S = O =, D = ) P (O =, D = ) = P (S = D = ) P (D = O = ) + P (S = O =, D = ) P (O =, D = ) = D + D = + 0 = 4 5

6 +5 (or )