Cathy Griffin has purchased a Mercedes Benz E63AMG. She puts its key into one of

Question

Cathy Griffin has purchased a Mercedes Benz E63AMG. She puts its key into one of 3 empty boxes: box 1, box 2, and box 3. She has closed all the boxes, so you do not know which box has the key. She will give you the new car if you select the box that contains the key. You select box 1 but do not open it. Of the remaining 2 boxes, she opens the box that does not have the key. She opens box 3. She gives you the option of replacing your box 1 with box 2. a. Calculate the probability that box 1 contains the key. b. Calculate the probability that box 2 contains the key. Several methods for calculating the above probabilities exist. One method uses a tree diagram with the following events. Ak = box k contains key Bk = Merkel opens box k At the root of the tree is the event that you select box 1.

Explanation / Answer

This an example of a typical Monty Hall problem:

The following table gives all the possibilities before the game which can happen:

Now clearly as we see from the above table if we choose Box 1. There is obviously a 1/3 probability that that box will have the key but afterwards as one of the box with no key is opened, if the person switches the box he wins a key in 2 out of 3 possibilities while if he does not switches the box he will win in only one of the 3 instances ( as shown inthe last 2 columns of the table) . Therefore as he has been given the option to switch the box, he should switch to box 2 because there is a 2/3 chance of him winning the key in case he switches and only 1/3 chance of winning the key in case he does not switch the box.

Box-1 Box-2 Box-3 Payoff on switching Payoff on no switching Key No key No Key 0 Key No Key Key No Key Key 0 No Key No Key Key Key 0

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