Problem 1 (10 points): Suppose a contestant on a game show is given the choice o

Question

Problem 1 (10 points): Suppose a contestant on a game show is given the choice of four closed doors and will win the prize behind the chosen door. Behind one of the doors is a car, behind the others, goats. The contestant chooses door 1. The host, who knows what's behind the doors opens door 2, which has a goat. He then offers to let the contestant choose between doors 3 or 4 instead of 1, if the contestant would like to do so. Would the contestant improve the chances of winning the car by choosing between doors 3 and 4 instead of 1? Given that the contestant originally chose 1 and the host revealed that the car is not behind 2, what is the conditional probability that the car is behind door 3? With the same given conditions, what is the conditional probability the the car is behind door 1? The answer to this problem depends on the assumptions. Assume that initially the car was equally likely to be behind any of the 4 doors, and the host always chooses to open one of the doors that was not chosen and does not contain the car, with equal probability for any of these 2 or 3 choices.

Explanation / Answer

Given that there are 4 doors , say D1, D2, D3, and D4. There is a car behind one door and goat behind the remaining 3 doors. Your chances of picking the right door (with the car) out of the 4 door is ¼ each .

Now the contestant chose D1. The host open Door 2 which has a goat. So , now the contestant is left with D 1, D 3 and D4.

Now the chances of choosing the right door between D2 and D3 is 1/3 each, whereas the chance of choosing D1 still remains the same i.e ¼.

Therefore we now have Chances of picking D1 is ¼, and D2 and D3 is 1/3 each which is greater . therefore the contestant much switch his choice of picking the D3 or D4. Switching sounds a pretty good idea as the contestant is more likely to win the car.

(Refrence: Monty Hall paradox)

Additional Information :

The general principle is to re-evaluate probabilities as new information is added. For example:

These are general cases, but the message is clear: more information means you re-evaluate your choices. The fatal flaw of the Monty Hall paradox is not taking Monty’s filtering into account, thinking the chances are the same before and after he filters the other doors.

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