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The Answer:
In the beginning of the problem the odds of finding gold behind the door you chose were 1 in 3, and the odds of the gold being behind one of the other two doors were 2 in 3. Those odds don't change, even though Monty gives you a hint. Kolmogorov's Axiom says that the odds of all the choices in a problem set must always add up to 100%. If the gold was hidden at random behind one of two doors, yes, the odds would be 50/50. But that's not what happened here. You and Monty agreed that the other two doors together offered a 2 out of 3 chance and, knowing what was behind each door, he told you which one of the pair not to pick. That doesn't move the gold. But if you thought the odds were 50/50 you are in good company. Even Paul Erdos, one of this century's great mathematicians, was in the 50/50 camp at first. When Marilyn Vos Savant published this puzzle and its answer in Parade Magazine, professors of mathematics scolded her in letters that are less about mathematics than about the tyranny of the tribe: "There is enough mathematical illiteracy in this country, and we don't need the world's highest IQ propagating more. Shame!" "I am in shock that after being corrected by at least three mathematicians, you still do not see your mistake." "If all those Ph.D.'s were wrong, the country would be in very serious trouble." The answer to this complex and controversial puzzle is completely contrary to our intuition, which is why it is so delightful. And confronting us with that cognitive dissonance--an idea contrary to belief, to public opinion and to self interest -- is one of the puzzle's gifts. The great lesson of science is to respect the evidence, wherever it leads. Kepler, for example, studied Tycho's data for years and saw only circular orbits until he finally realized that the paths were eliptical. "It is the theory that allows us to see the evidence." (Einstein) Think of the Monty Hall puzzle as presenting two tasks: calculate the probability, and learn to see and understand evidence in conflict with "common sense". |
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Here's a tough one you can do yourself. J. Paul Getty, one of the most successful oil men of his time, was once asked how he managed to strike oil more often than any of his competitors. "When I hire a geologist," Getty said, "I ask him if he has ever struck oil before. If he says yes I hire him. If he says no, I don't." But another oil company has a more complex process. Only 10% of the geologists they interview know how to find oil. The company can spot those engineers correctly 80% of the time, and they hire everyone they think is qualified. Of 100 geologists they interview, how many will they hire, and what percentage of those can really do the job? |
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