Monty Hall Problem - Page 2
Some people say, well there are just two doors left, and each has an equal chance of being right, so now the contestant has a 50% chance of picking the right door. Right? Well, no, actually. Now it IS true that if at this point - with just the two doors left - you brought a new contestant into the studio, who had seen nothing of what had gone before, and gave them the choice of the two doors, THEY would have a 50% chance of picking the winning door. But that is not the situation. Our contestant picked out a door from the original line-up of 3 doors, and that original choice of door had a 1 in 3 chance of being right, and it STILL has a 1 in 3 chance of being right. Whereas if they switch, they will have a 2 in 3 chance of being right.
One way of putting it is you are choosing between your original choice of door and BOTH the other doors, but with the host having conveniently shown you which of the other two doors NOT to choose - so you are really choosing BOTH the other two doors, which gives you a 2 in 3 chance of being right.
Another way of putting it is to think of a situation with 100 doors, but only 1 prize, and 99 goats. Your initial choice of door has a 1 in 100 chance of being right. The host then runs up and down opening doors all over the place, the stage is full of goats, and finally there are only two doors left. Your original choice of door, and one other. I hope you will agree that your original choice still has a 1 in 100 chance of being right, and the one remaining other door has a 99 in 100 chance of being right?
Some years ago, when the writer and journalist Marilyn vos Savant quoted this puzzle in a US newspaper article, she received over 10,000 letters mostly telling her she was wrong. One was from Robert Sachs, a professor of mathematics at George Mason University in Fairfax, Va. who said "As a professional mathematician, I'm very concerned with the general public's lack of mathematical skills. Please help by confessing your error and, in the future, being more careful." However a week later and Dr. Sachs wrote her another letter telling her that "after removing my foot from my mouth I'm now eating humble pie. I vowed as penance to answer all the people who wrote to castigate me. It's been an intense professional embarrassment."
Still not convinced? See the next page...