Hack 50. Avoid the Zonk

On the TV game show Let's Make a Deal, contestants often had to choose between three curtains. For these sorts of situations, there is a statistical strategy that will help you to win the Buick instead of the lifetime supply of Rice-A-Roni.

Imagine, if you will, that you are traveling with your Uncle Frank through an uncharted region of Tonganoxie, Kansas. You come to a fork in the road that branches out into three possible paths: A, B, and C. You don't know which will lead you to your destination, the fabled world's largest ball of twine (in Cawker City, Kansas). An old prospector is resting with his burro at the crossroads.

"Say, old timer," you say, "which road leads to the world's largest ball of twine?"

"Well," says he, "I know, but I won't tell you. What I will do, though, is tell you that one road is the correct road. Two are wrong and lead to certain disaster (or at least poorly maintained restrooms). Go ahead and take your pick, city slicker. As you drive off, look back at me. I won't signal whether you are right or wrong, but I will point at one of the other two roads. The one I point at will be a wrong road. You still won't know for sure whether you guessed right or not, of course, but I guarantee that I'll point at one of the two roads you are not on and it will be a wrong road."

You accept the strange man's offer (what choice do you really have?) and you ask Uncle Frank, the experienced gambler among you, to pick a road. He does so randomly and you head off optimistically down one of the three pathslet's say A. As you look back, the kindly prospector points to one of the other roadslet's say B. Immediately, you slam on the brakes and back the car up. Over the objections of Uncle Frank, you head down the remaining road, C, with the peddle to the metal, fairly confident that you are now on the right path.

Crazy, are you? Suffering from white-line fever? No, you've just applied the statistical solution to what is known as the Monty Hall problem and chosen the road among the three that has the greatest chance of being correct. Hard to believe? Read on, my friend, and prepare to win riches beyond your wildest dreams.

The best strategy in this case is so counterintuitive and downright weird that the world's smartest people have disagreed aggressively about whether it even really is the best strategy. But believe meit is.

The Monty Hall Problem and Game Show Strategy

In our example with the three roads and the prospector, there is, in fact, a two-thirds (about 67 percent) chance that C is the correct road. To apply this odd strategy to a more realistic situation, think of contestants on game shows or gamblers in any game in which prizes are hidden in boxes or behind doors. As typically discussed among game show theorists and cranky statisticians, the problem is presented as a fairly common actual situation on the game show Let's Make a Deal (which had its heyday in the 1960s and 1970s), but it is a situation still seen today in TV game shows. The host of Let's Make a Deal was Monty Hall, so the problem carries his name.

As a game show scenario, the problem goes like this. Monty presents to you three curtains. He knows what is behind each curtain. He explains that behind one of the curtains is a brand-new car. The other two curtains hide worthless prizes, what Monty used to call zonks. (Zonks were often something like a donkey or a giant rocking chair, something that wouldn't be of any real use.) He lets you pick a curtain, and you will win whatever is behind it. Let's say you pick curtain A. He then opens one of the unchosen curtainsB, for exampleto show you that it has a zonk behind it. He then offers to let you trade your original choice for the remaining curtain, C. Should you switch?

As with the three roads problem, the answer is yes, you should switch. The answer just never seems right the first time one hears it. But, if you want to increase your odds of winning the car, you should now switch.

Why You Should Always Switch

Think of the probability of you guessing the correct curtain. Let's assume that it is a random guessnone of this "I notice that one curtain moved, so I figured there was a donkey behind it" stuff.

Three curtains, with only one curtain being a winner, means there is a 1 out of 3 chance that you will guess right and win the car. That's about 33 percent. On that first guess, with no additional information, you are likely to be wrong; in fact, you have a 2 out of 3 chance of being wrong. In other words, there is about a 67 percent chance that the car is somewhere behind the two curtains you did not pick.

Once you know that one of those other two curtains does not have the car, that doesn't change the original probability that the car is 67 percent likely to be somewhere behind those two unselected curtains. Remember, Monty will always have a wrong curtain he can open, no matter which one you choose. The 67 percent chance that the car is behind B or C remains true, even after B is revealed to not be hiding the car. The 67 percent likelihood now transfers to curtain C. That's why you should always switch to the other curtain.

If you were given the option of swapping your pick of one curtain for both the other two curtains, you'd switch in a second wouldn't you? That's essentially what is offered in the Monty Hall problem.

Some figures might be necessary to persuade your inner skeptic. Look at Table 5-1, which shows the probability breakdown for the three options at the start of the game. You have a one-third chance of guessing the winning curtain and a two-thirds chance of picking a nonwinning curtain.

Table Probability of car's location at start of game

Curtain A	Curtain B	Curtain C
33.33 percent	33.33 percent	33.33 percent

Table 5-2 shows the same probabilities grouped in a different way, but it hasn't changed any of the parameters of the problem.

Table Restated probability of car's location at start of game

Curtain A	Curtain B or Curtain C
33.33 percent	66.66 percent

Table 5-3 shows the probabilities after Monty reveals one of the nonchosen curtains (Curtain B) to be a nonwinner. The 67 percent likelihood now transfers to curtain C.

Table Probability of car's location after curtain B is opened

Curtain A	Curtain B	Curtain C
33.33 percent	0.00 percent	66.66 percent

In any situation like this, you should switch. You might be wrong, of course, but you have a better shot of winning that car or whatever other prize you are playing for if you accept any offers to switch. This is always the best strategy, if a few criteria are met:

• The host knows what is behind each curtain.

• The host reveals one of the unchosen curtains and the prize is not behind it.

• Your original choice was random.

The Controversy

The Monty Hall problem and the general game show strategy that resulted was first introduced to the masses in 1991 by Marilyn Vos Savant, a columnist for Parade Magazine. Because she is known for being a "high IQ genius," Vos Savant answered questions from readers, sometimes of a brain teaser nature. Someone sent in the problem as I've described it, and she published the answer I have given here.

Apparently, she received many letters, some angry, from statisticians, philosophers, and such claiming that she got it wrong. In scholarly journals, there were even published debates about whether her answer was correct. My read of the debate is that it turned out that most of the arguments centered on a key ingredient of the question: Monty knows what is behind each door, so when he opens that first curtain, he knows it will be a zonk.Otherwise, the reveal does not count as new information and the answer Vos Savant gave does become debatable. Most of the critics of her answer missed that part of the original published question.

Don't be too concerned if the correctness of this solution isn't immediately apparent. Really smart people often first view the new odds as being 50/50 between the two unopened curtains and, therefore, it doesn't matter if you switch. The key to remember, though, is that your original chance of picking the correct door, 33.3 percent, cannot change no matter what happens after you make your choice. Even experts sometimes disagree about the best way to view this question. Even people as wise as the old prospector you met out in Tonganoxie that started our discussion don't always know the right answer to the Monty Hall problem. How do you think he won that burro?