Problem 186 Solution

The key to this problem is that the host is predestined to open a door with a goat. He knows which door has the car so regardless of which door the player picks he always can reveal a goat behind another door.

Let's assume that the prize is behind door 1. Following are what would happen if the player had a strategy of not switching.

• Player picks door 1 --> player wins
• Player picks door 2 --> player loses
• Player picks door 3 --> player loses

Following are what would happen if the player had a strategy of switching.

• Player picks door 1 --> Host reveals goat behind door 2 or 3 --> player switches to other door --> player loses
• Player picks door 2 --> Host reveals goat behind door 3 --> player switches to door 1 --> player wins
• Player picks door 3 --> Host reveals goat behind door 2 --> player switches to door 1 --> player wins

So by not switching the player has 1/3 chance of winning. By switching the player has a 2/3 chance of winning. So the player should definitely switch.

The plain simple English reason the probability of winning increases to 66.67% by switching, lays in the fact that the host always reveals a goat. If the host didn't know which door had the car, then the probability of having a win would go up to 50% after revealing one of the goats. Such is the case on the show "Deal or no Deal." On that show, the host does not know where the million dollar case is. So, as cases are eliminated that do not have the million dollars, the probability increases that every remaining case has it, equally. When there are only two cases left and two prizes, each case has a 50% chance of each prize.

Links