About a year ago I stumbled across a seemingly very simple probability question that I thought might provide some entertainment here. I would like to request that those with a Master's or higher in math (or related sciences) wait 24 hours from this post time before replying to give others a chance. Bragging rights for the first person to post the correct probability in percentage answer with an explanation of 10 lines or less. No googling allowed.
THE PROBLEM
Those of us old enough to remember watching the popular and cheesy 1970's game show, Let's Make A Deal, will be able to relate to the situation easily. In the game show, the winner at the end of the show was shown 3 great big doors (the size of a bay door at a garage) with the numbers 1, 2, and 3 painted on them. One of these doors contained a valuable prize (like a new car or a vacation package), the other contained a very modest prize (like a stereo - remember it was the 70's) and the other door had a gag prize (like a goat or 50 boxes of macaroni and cheese). The contestant had to choose one of the 3 doors to reveal their prize. Often, but not always, before opening the door selected, the host would ask the person if they wanted to switch their pick. Sometimes to entice them to switch, the host would open the door with the modest prize and offer that to them right now in exchange for forfeiting their pick. With 2 doors to choose from, it would seem that there was now a 50/50 chance of getting the car (or the goat if you happen to be a goat farmer looking to expand your herd
). But is it a 50/50 chance? Here is the question again in its correct form that removes any ambiguity:
A thoroughly honest game-show host has placed a car behind one of three doors. There is a goat behind each of the other doors. You have no prior knowledge that allows you to distinguish among the doors. "First you point toward a door," he says. "Then I'll open one of the other doors to reveal a goat. After I've shown you the goat, you make your final choice whether to stick with your initial choice of doors, or to switch to the remaining door. You win whatever is behind the door." You begin by pointing to door number 1. The host shows you that door number 3 has a goat.
Do the player's chances of getting the car increase by switching to Door 2?
The standard analysis of the problem also assumes that the host opens one of the remaining two doors randomly if the player initially picked the car.
THE PROBLEM
Those of us old enough to remember watching the popular and cheesy 1970's game show, Let's Make A Deal, will be able to relate to the situation easily. In the game show, the winner at the end of the show was shown 3 great big doors (the size of a bay door at a garage) with the numbers 1, 2, and 3 painted on them. One of these doors contained a valuable prize (like a new car or a vacation package), the other contained a very modest prize (like a stereo - remember it was the 70's) and the other door had a gag prize (like a goat or 50 boxes of macaroni and cheese). The contestant had to choose one of the 3 doors to reveal their prize. Often, but not always, before opening the door selected, the host would ask the person if they wanted to switch their pick. Sometimes to entice them to switch, the host would open the door with the modest prize and offer that to them right now in exchange for forfeiting their pick. With 2 doors to choose from, it would seem that there was now a 50/50 chance of getting the car (or the goat if you happen to be a goat farmer looking to expand your herd
). But is it a 50/50 chance? Here is the question again in its correct form that removes any ambiguity:A thoroughly honest game-show host has placed a car behind one of three doors. There is a goat behind each of the other doors. You have no prior knowledge that allows you to distinguish among the doors. "First you point toward a door," he says. "Then I'll open one of the other doors to reveal a goat. After I've shown you the goat, you make your final choice whether to stick with your initial choice of doors, or to switch to the remaining door. You win whatever is behind the door." You begin by pointing to door number 1. The host shows you that door number 3 has a goat.
Do the player's chances of getting the car increase by switching to Door 2?
The standard analysis of the problem also assumes that the host opens one of the remaining two doors randomly if the player initially picked the car.
