Back to the task of persuading you that you should switch. I tried putting in a table to indicate the virtues of switching, but it didn't look very good. So, if you will click Download monty_hall_table.doc, you will get access to the table in Word format.
Notice that the table contains all nine possibilities and that if you always switch from the door that you originally chose, you will win six times.
Here's one more variation that is due to one of my students, Eric Berger. Imagine that I have hidden a $10,000 bill in a book in the library of the University of Illinois College of Law. I know exactly where the book containing the money is. Assume that there are 10,000 books in the library. I invite you to choose one of the books. If it is the book containing the money, you get to keep the money.
Your chances of choosing the right book are 1/10,000 -- not very good. Now suppose that you have chosen a book. We put it on the table, unopened. It might contain the $10,000 bill; it might not.
Imagine that I now take another book from the shelves and open it to reveal that it does not contain the bill, and we put that book aside. I ask if you would like to switch from the book you originally chose to another unopened book in the library. (There are now 9,998 of those.) And you might well say, "Why should I?"
Suppose that I keep taking books off the shelves, showing you that each of them does not contain the $10,000, and then setting them aside. After an interminable time there are only two books left -- the one you originally chose and one other. Should you switch? Sure. You might argue that after you made your original choice, the probability that the $10,000 is hidden in one of the other 9,999 books in the library is 9,999/10,000 (almost 1). I have now opened 9,998 of those books. All that remain are the book you originally chose and one other unopened book. Isn't it fairly clear that all of the 9,999/10,000 probability that was distributed across the books that you did not originally choose now rests on that other book?
One or more posts on this in the near future.
TSU
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