Monty Hall, Goats, And Innumeracy
Steven Dutch, Natural and Applied Sciences, Universityof Wisconsin - Green Bay
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There are some puzzles where the "correct" answer becomes so ingrained in popular culture that it becomes almost impossible to convince people the answer is wrong.
For example, my father used to be fond of a riddle that went "Brothers and sisters I have none, but this man's father is my father's son." He was absolutely convinced the answer was that the man was looking at a picture of himself. So I even drew him diagrams like the one below:
| My Father | |
| This Man's Father is | My Father's Son (Me) |
| This Man | My Son |
The popular answer is wrong. The riddle actually refers to the man'sson. Do you think I could make my father see it? His response was to repeat the riddle, emphasizing different parts of it, as if saying it louder somehow overcame logic.
Pretty much the same thing has happened with the infamous "goat problem" or "Monty Hall problem." You are confronted with three doors. Behind two of the doors is a goat and behind the other is a car. You pick a door. Monty opens one of the other doors and shows you a goat. Now, should you switch doors or not? Many people intuitively say it doesn't make any difference but the answer, most everyone who writes about this subject assures is, is yes - you have a 2/3 probability of winning if you switch. We even have the word of Marilyn vos Savant, world renowned filler-in of bubbles on IQ test sheets.
What you want to do depends on the car. If it's a former East German Trabant, a name that ironically means "treasure," you're better off with the goat.
I once had a very frustrating exchange with a statistician where I challenged him to enumerate all the possible outcomes and show that the probability of winning was 2/3. Considering that enumerating possible outcomes is what you do in the first lecture in intro statistics to show the probability of rolling a 7 with dice or flipping 5 heads in a row, you'd think he's get it, a statistician, of all people. Anyway, here is the complete matrix:
Actually, it's not the complete matrix, because I need five more for all the possible arrangements behind the doors: C12, C21, 1C2, 12C, 2C1 and 21C. I hope you can see they'll all come out the same way. There are 36 outcomes for each ordering: three doors to choose from times three initial choices times two ways Monty can pick the wrong door times two final options: hold or change.
Basically, you have a 50% chance of winning if you change, and a 50% chance of losing. But the odds of picking the door with the car are 1/3 and picking a goat are 2/3. So there are twice as many favorable outcomes from switching if you picked the wrong door than losing if you had picked the right door. The chances work out to 6 hold and win, 6 hold and lose, 12 change and win, 12 change and lose. Note that the chances of winning this game are 50%.
The Airline Passenger
A related problem, described by Martin Gardner in Scientific American when he discussed the goat problem, is this: You're on an airliner and your seatmate tells you he has two children. A little while later, he mentions his son. What are the odds that the other child is a girl?
Well, normally the odds of having one of two children be a girl is 3/4. There are four possibilities for two children: BB, BG, GB and GG. However, by mentioning his son, the passenger has eliminated GG. That leaves only BB, BG and GB, so that the odds of the other child being a girl drop to 2/3.
However, just like with the goat problem, not only does the probability of the second child being a girl change, so does the probability that the second child is a boy. Before you knew your seatmate had a son, the probability of his having two sons was 1/4. After he tells you he has a son, he eliminates possibility GG and the probability of the second child being a boy rises to 1/3 - it is not locked in permanently at 1/4. There's only one way he can have a second son, but two ways he can have a daughter, so the odds of a daughter are twice as great.
Here's how Marilyn vos Savant put the problem in her column in 1990:
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. You pick a door, say No. 1 [but the door is not opened], and the host, who knows what's behind the doors, opens another door, say No. 3, which has a goat. He then says to you, "Do you want to pick door No. 2?" Is it to your advantage to switch your choice?
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