6. Problem 3.5.16 from Yates and Goodman In a TV game show, there are three identical-looking suitcases. The first suitcase has 3 dollars, the second has 30 dollars and the third has 300 dollars. You start the game by randomly choosing a suitcase. Between the two unchosen suitcases, the game show host opens the suitcase with more money. The host then asks you if you want to keep your suitcase or switch to the other remaining suitcase. After you make your decision, you open your suitcase and keep the D dollars inside. Should you switch suitcases? To answer this question, solve the following subproblems and use the following notation: Ci is the event that you first choose the suitcase with i dollars. Oi denotes the event that the host opens a suitcase with i dollars. In addition, you may wish to go back and review the Monty Hall problem in Example 2.4. (a) Suppose you never switch; you always stick with your original choice. Use a tree diagram to find the PMF Pd(d) and expected value E[D]. (b) Suppose you always switch. Use a tree diagram to find the PMF PD(d) and expected value E[D]. (c) Perhaps your rule for switching should depend on how many dollars are in the suitcase that the host opens? What is the optimal strategy to maximize E[D]? Hint: Consider making a random decision; if the host opens a suitcase with i dollars, let di denote the probability that you switch. 6. Problem 3.5.16 from Yates and Goodman In a TV game show, there are three identical-looking suitcases. The first suitcase has 3 dollars, the second has 30 dollars and the third has 300 dollars. You start the game by randomly choosing a suitcase. Between the two unchosen suitcases, the game show host opens the suitcase with more money. The host then asks you if you want to keep your suitcase or switch to the other remaining suitcase. After you make your decision, you open your suitcase and keep the D dollars inside. Should you switch suitcases? To answer this question, solve the following subproblems and use the following notation: Ci is the event that you first choose the suitcase with i dollars. Oi denotes the event that the host opens a suitcase with i dollars. In addition, you may wish to go back and review the Monty Hall problem in Example 2.4. (a) Suppose you never switch; you always stick with your original choice. Use a tree diagram to find the PMF Pd(d) and expected value E[D]. (b) Suppose you always switch. Use a tree diagram to find the PMF PD(d) and expected value E[D]. (c) Perhaps your rule for switching should depend on how many dollars are in the suitcase that the host opens? What is the optimal strategy to maximize E[D]? Hint: Consider making a random decision; if the host opens a suitcase with i dollars, let di denote the probability that you switch
