Question

In: Economics

consider the Monte Hall problem as discussed in lecture. Recall that a game show host(Monte Hall)...

consider the Monte Hall problem as discussed in lecture. Recall that a game show host(Monte Hall) gives a contest a chance to choose from three doors of which one is a new car and the other two are goats. After the contestant chooses a door, the game show host who knows what is behind all of the doors decides to open another door behind which a goat sits. The games show host. then, offers the contestant an opportunity to switch doors. For the purposes of this question, note the following:

1. The contestant initially selects  door B

2.Game show host (Monte Hall) open door C behind which a goat sits

also note the following notation for events

1. Car A= the car is actually behind the door A

2. car B= the car is actually behind the door B

3. car C= the car is actually behind the door C

4. A= the host reveals a goat behind Door A

5.B=the host reveals a goat to be behind Door B

6.C= the host reveals a goat to be behind Door C

Given that the host reveals a goat behind Door C, what is the (conditional)probability that the car is actually behind door A [P(Car A| C)]?

Solutions

Expert Solution

The contestant selects Door B and there is a probability of the car being behind any of the doors of 1/3.

However, the probability of a goat behind each of the doors is 2/3

Now the door C is opened and a Goat is revealed.

Initial Choice Prize Behind Door Opened Switch Don't Switch
A A B or C Lose Win
A B C Win Lose
A C B Win Lose
B A C Win Lose
B B A or C Lose Win
B C A Win Lose
C A B Win Lose
C B A Win Lose
C C A Lose Win

From the above table we see that the number of cases when the contestant chooses door B and door C is opened is 2.

Out of those 2 cases, the car is behind A in only of the cases.

So the probability of the car being behind A while not being behind C is 1/2.

In general, if the contestant switches, he wins 6 out of 9 times or 66.67% times and if the contestant does not switch he wins 33.33% times as per the above table.

If you found this helpful, please rate it so that I can have higher earnings at no extra cost to you. This will motivate me to write more.

Rahul Sunny answered 2 years ago
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