The Goat and the Car: A Probability Puzzler

(Also known as the Monty Hall Paradox)

This is mind-teaser involving simple statistics which even many PhDs in physics and math get wrong. It's become associated with Marilyn Vos Savant, who gave the correct answer in her popular syndicated question and answer column. Many people who read her answer didn't believe it, and wrote the her or the publications' editors.

If you've never heard of Marilyn Vos Savant, She's listed in the Guinness Book of World Records for having the highest recorded IQ. She has written several books.

She is not the only one who has discussed this in published writing -- Jeffrey D. Kooistra discussed the same puzzle in Analog Science Fiction / Science Fact, one of my favorite magazines. A computer programming magazine had an article on it. It pops up repeatedly in Mensa's various publications. And more... see my references page

Every time this puzzle is explained in magazines, in internet newsgroups, or anywhere else, it always seems like there's a small group of clear-thinking individuals arguing against the great ignorant (though well-educated) hordes. "Ignorant though well-educated"? Yes. No matter how smart some people think they are, there are always mental traps they can stumble into. Although, to be fair, there are several assumptions that are reasonable to make, but may be rejected or modified. A real-life game show is show biz and psychology, not pure mathematics.

The Puzzle

You are the contestant on a TV game show. This game show is exactly like a certain one that really was on the air, in fact. You may remember it. You stand nervously on stage under hot bright lights. The charming host points toward three big doors, labelled A, B and C. "One of these doors hides a shiny brand new bright red Chrysler Neon! Behind the other two doors are -- dirty barnyard animals! Which door do you pick?"

Well, you can't see through the doors, and the goats are being very quiet. Which door will you pick? Hmmm.... How 'bout... "That one! door C!" you announce with trepidation and anxious hope.

"Nice choice, but before we open it, I'm going to reveal what's behind one of the other doors - door A! Look -- it's..." The audience holds their breaths as door A swiftly raises. BAHAHAHA! "...a smelly goat!"

"Now I bet you're glad you didn't choose door A!" Chuckles from the audience fill the air briefly while the the host puts on a serious look. "Now I like to give everyone a chance to change their mind. You've chosen door C. We both know that there's a brand new car behind either door B or door C. Are you excited?" Yeah, yeah, but, like, can we get on with things now? Is what you'd like to say. "I'm giving you the chance now to change your mind - to switch to door B, if you want.

"Do you want to stay with door C, or switch to door B?"

This is the puzzle - does it matter if you switch? Since there are two doors, and one has a car and the other a dirty animal, isn't it 50-50? Seems like you can switch or not with equal statistical outcomes. Does it matter? Is there some trick? Maybe the hot studio lights are getting to your brain?

You may want to read the original statement of the problem, as printed in Parade magazine (a Sunday supplement provided by many newspapers.)


Nearly everyone who reads this puzzle thinks, at first, that is really is 50-50. But the surprising answer is: You are better off switching doors! You are twice as likely to win by switching than by staying with your first choice.

You can read a theoretical explanation or if you prefer hands-on direct experience, try an experiment. If you're not convinced by these arguments, here is a very short and perfectly logical one by Bob Metcalfe. There's a longer one on the Cut-the-Knot web site. There are many more explanations on other web sites listed on the references page.

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