21

I watched '21' yesterday and I found one questioned posed quite interesting. The Monty Hall problem:

"Suppose you're on a game show and you're given the choice of three doors. Behind one door is a car; behind the others, goats. The car and the goats were placed randomly behind the doors before the show. The rules of the game show are as follows: After you have chosen a door, the door remains closed for the time being. The game show host, Monty Hall, who knows what is behind the doors, now has to open one of the two remaining doors, and the door he opens must have a goat behind it. If both remaining doors have goats behind them, he chooses one randomly. After Monty Hall opens a door with a goat, he will ask you to decide whether you want to stay with your first choice or to switch to the last remaining door. Imagine that you chose Door 1 and the host opens Door 3, which has a goat. He then asks you "Do you want to switch to Door Number 2?" Is it to your advantage to change your choice?"


Without giving much thought, one would simply answer that there is no probability difference in switching or remaining. People would ordinarily choose to stay with the same door as they assume that the host is trying to influence their decision. They would be wrong.



As the matrix above shows, in 2/3 of the possible scenarios, switching leads to the car. Hence in the situation described, it is always more beneficial to switch - probability wise anyway. Which door is chosen first is inconsequential.