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The Monty Hall Problem 

click to enlarge Six-game matrix exhausts all possibilities for when you pick door 1. If Monty predictably opens a "goat" door, switching (games 1, 2 and 3) increases your chance of winning the car to 2/3 compared with 1/3 if you stay (games 4, 5 and 6).
  • Six-game matrix exhausts all possibilities for when you pick door 1. If Monty predictably opens a "goat" door, switching (games 1, 2 and 3) increases your chance of winning the car to 2/3 compared with 1/3 if you stay (games 4, 5 and 6).

So much ink has been spilled over the "Monty Hall Problem" that a little more won't make any difference (first fallacy!). The "paradox" (second fallacy) got a huge play 20 years ago in Marilyn von Savant's Parade magazine column, and resurfaces at regular intervals. Here's how Marilyn posed it.

You're on a Monty Hall's Let's Make a Deal TV show, and he gives you the choice of three doors: behind one door is a car; behind the other two, goats. You pick a door, say No. 1, and the host, who knows what's behind the doors, opens one of the other doors -- say, No. 3 -- revealing a goat. He then says to you, "Do you want to switch your choice and pick door No. 2?" Is it to your advantage to switch? The gut reaction of most people is no -- there's a 50/50 chance that either of the two closed doors hides a car, so whether you switch or not is immaterial.

When Marilyn insisted that switching does improve your chances of winning the car, about 10,000 of readers took her to task, an unprecedented response to her column. "How could switching doors possibly make a difference?" they asked. "You either picked the right door or you didn't, and the fact that one of the doors is now open doesn't change that!"

Wrong, said Marilyn, reasoning like this: The contestant picks (say) door 1, after which Monty gives valuable information to the contestant by eliminating one of the two "goat" doors. Monty didn't change the 1/3 chance of door 1 hiding a car. Since there's now only one other closed door, the chance of the car being behind that door is 2/3, because the chances add up to 1 (representing the certainty that a car is behind one of the two closed doors). Switching therefore increases your odds from 1/3 to 2/3.

The debate rolled on through several subsequent issues of Parade. How come? Either 10,000 readers were right or Marilyn was right! The trouble lay in the ambiguous statement of the problem: Marilyn failed to specify that Monty never opens the "car" door, he always deliberately opens a "goat" door. (If Monty was clueless and just happened to open a "goat" door, there's no advantage in switching -- but no disadvantage either -- so you still might as well switch.)

There are many variations: You suspect Monty is malicious, and that he offers the chance to switch more often if you picked the car door (don't switch!). Or you think Monty is angelic, and offers the option to switch only when you choose incorrectly (switch!). Or you've already got a good car -- you really want to win a goat!

Barry Evans (barryevans9@yahoo.com) is trying to improve his odds of winning eight-ball at Steve and Dave's bar in Old Town Eureka, where he lives (in Old Town, not in the bar).

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