3. At end of a television game show, the contestant is presented with three clos

Question

3. At end of a television game show, the contestant is presented with three closed doors. Behind one of the doors there is a car, and behind each of the other two doors there is a goat. He gets to choose a door and keep whatever is behind it.

After he makes his choice but before the door is opened, the show’s host opens one of the other two doors, which he knows conceals a goat. The host then asks him whether he wants to switch his choice to the remaining closed door or stick to his original choice.

(a)       Should he switch or not?

Hint. A contestant who decides before the show to stick with his original choice wins the car if and only if his original choice was correct. A contestant who decides before the show to switch wins the car if and only if his original choice was wrong. By how much does switching increase his chance of winning?

(b)      Suppose now that there are 1000 doors, 1 car and 999 goats. The contestant chooses 1 door. The host then shows 998 doors with goats. By how much does switching now increase his chance of winning? Notice that testing our answer in extreme cases helps us sort out our intuition.

(c)       Consider again the original situation with three doors. The contestant chooses a door. Suppose now that the show’s host does not know what’s behind each door. However, the show’s host opens one of the other two doors, which happens to reveal a goat. Should the contestant switch or not?

3. At end of a television game show, the contestant is presented with three closed doors. Behind one of the doors there is a car, and behind each of the other two doors there is a goat. He gets to choose a door and keep whatever is behind it. After he makes his choice but before the door is opened, the show's host opens one of the other two doors, which he knows conceals a goat. The host then asks him whether he wants to switch his choice to the remaining closed door or stick to his original choice. (a) Should he switch or not? Hint. A contestant who decides before the show to stick with his original choice wins the car if and only if his original choice was correct. A contestant who decides before the show to switch wins the car if and only if his original choice was wrong. Bv how much does switching increase his chance of winning? (b) Suppose now that there are 1000 doors, 1 car and 999 goats. The contestant chooses 1 door. The host then shows 998 doors with goats. By how much does switching now increase his chance of winning? Notice that testing our answer in extreme cases helps us sort out our intuition (c) Consider again the original situation with three doors. The contestant chooses a door. Suppose now that the show's host does not know what's behind each door. However, the show's host opens one of the other two doors, which happens to reveal a goat. Should the contestant switch or not

Explanation / Answer

Case a)

No. of doors : 3

Since we only need to calculate how much better is switching the doors as compared to not switching to win in the game hence we are considering only cases where contestant wins.

1. To win and swtich it is clearly mentioned that contestant should be wrong in orignal choice.

Probablilty of choosing a wrong door , then switching later and winning : 2/3 (as 2 doors have goats) =0.66

2. To win without switching doors the contestant has to be correct in orignal choice

Probability of choosing a right door, and not switching later to win: 1/3 = 0.33

Hence in case (a) switching improves chances by 0.33 (0.66 - 0.33) or by 50% [(0.66-0.33)*100/0.66].

Case b) Similar to case we will consider only those cases in which contestant wins:

1. Probability to win by switching the doors(To win via switch the contestant always has to choose a wrong door): 999/1000 = 0.999

2. Probability to win by not switching the doors (To win contestant will have to choose the right door): 1/1000 = 0.001

Hence the chances by switching are now improved by 0.998 or 99.9%[(0.999-0.001)*100/0.999].

Case c) Assuming same case: The contestant has to always win the contestant

1. To win with switch in this case :

1.1 Probability of host choosing either goat's door or car door: 0.5 (After contestant chooses his door host can choose from only 2 doors)

1.2 Probability of contestant making a wrong guess : 0.66

Total : 0.5*0.66 = 0.33

2. To win without switching in this case :

2.1 Probability of host not choosing a car: 1 (As we know he will choose a Goat's door because contestant is going to win and not swtich)

2.2 Probability of contestant choosing the right door : 0.33

Total : 1 * 0.33 = 0.33

Hence the probability of winning with either with switching and not - switching remains the same.

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