Tuesday, March 3, 2009

Behind the Door

Yesterday I posed a probability question taken from the book I'm currently reading. Assuming that the students were lying about having a flat tire, success on their makeup exam depends on both of them naming the same tire on the car as being the one that went flat. What are the odds of that happening?

The author said most people would reason that each student had four tires from which to choose, and therefore the odds were 1/16 that they would both name the same tire (each of the four tires had an equal chance of being named by student A, thus 1/4, and same for student B, 1/4. 1/4 x 1/4 = 1/16.

But that's not correct. The correct answer is that there is a 1 in 4 probability (or the odds are 1 in 4) that both students will pick the same tire as the flat one.

Once student A has made his choice, be it the front driver or passenger side, or the rear driver or passenger side, then student B has a one in four chance of choosing the same tire student A chose. So the odds are 1/4.

Comprende? If the above is confusing to you, wait until you ponder this one, also from the same book.

Remember the old game show Let's Make a Deal? And how host Monty Hall would often offer contestants their choice of door number 1, door number 2, or door number 3? Behind one of the doors was a truly great prize, like say, tickets to see James Taylor. Behind the other doors were things that really stunk, like having to endure listening to Nancy Pelosi, or being forced to eat dinner with one of our California legislators.

Lets say you are the lucky contestant who gets to choose one of the doors. So you make a choice.

Naturally Monty is not going to show you what is behind your chosen door first. Since he knows what is behind all three of the doors, he will show you one of the two doors you didn't choose, just to make it more interesting.

He opens one of the other doors and there stands Nancy Pelosi. Whew, you managed to avoid her! She's too busy to pay any attention to you anyway and off she goes in her private jet to a conference. So that means that the good prize is either behind the door you chose, or the other door that remains closed.

If the good prize was behind the other door you didn't choose, of course Monty was not going to show that one to you and ruin the ending. Then he asks you if you would like to change your choice and switch it to the other door that is closed, instead of the door you originally chose.

Should you do it? Are the odds better for you if you switch, or if you don't switch?

And of course I am not going to tell you the answer today.. you'll have to wait to see what is behind the next blog post..

No comments: