He hasn’t shot an episode of Let’s Make a Deal for decades, but Monty Hall’s name still graces a statistical brouhaha from the early 1990s, and the drama he cultivated on air holds enduring lessons for all of us.
You probably remember how it worked: Monty would choose a member of the audience and then offer him a choice of three doors. One door concealed a fabulous prize: a load of cash or a car. The other two doors hid dud prizes, or “zonks” — disappointments like a goat or a pair of old shoes. So the chances of winning were pretty good — 1 in 3 — and definitely better than a scratch-off lottery ticket or slot machines. But here is where the wrinkle comes in: After you chose a door, Monty would typically open one of the other two doors, revealing a zonk. He would then give you a chance to switch from your first choice to the other unopened door.
Casual statisticians — that is, normal people — assume that chances are even (1 in 2) at this stage. So it becomes a question of “sticking to your guns” or having a wandering eye and abandoning your first choice for another.
In 1990, a columnist in Parade magazine, Marilyn vos Savant, wrote an article explaining why it always makes sense to switch from your initial choice. Sticking to your guns, she tried to explain, is, statistically, a failing strategy. This position earned her a heap of vitriol. Here is John Tierney writing about the affair in The New York Times in 1991:
Since she gave her answer, Ms. vos Savant estimates she has received 10,000 letters, the great majority disagreeing with her. The most vehement criticism has come from mathematicians and scientists, who have alternated between gloating at her ("You are the goat!") and lamenting the nation's innumeracy.
Her answer — that the contestant should switch doors — has been debated in the halls of the Central Intelligence Agency and the barracks of fighter pilots in the Persian Gulf. It has been analyzed by mathematicians at the Massachusetts Institute of Technology and computer programmers at Los Alamos National Laboratory in New Mexico. It has been tested in classes from second grade to graduate level at more than 1,000 schools across the country.
There are some excellent resources for understanding why, as Mr. Tierney put it, all these Ph.D.s in math were “dead wrong.” This clear explanation from the Khan Academy will set you straight in seven minutes:
If you don’t have seven minutes, let me explain. Say you choose door #1 and then Monty shows you a goat behind door #3. If you switch to door #2 you DOUBLE your chances of winning the car over staying with door #1. Why? Because the host revealed more than you think when he showed you door #3. If door #1 concealed the car, then the host had a choice of two doors to open. But chances of that are only 1 in 3. This means chances are 2 in 3 that your initial choice was a zonk and there was only one other zonk he could reveal. So once that zonk is in the open, door #2 gives you a 2 in 3 chance of winning. Stick with door #1 and your chances of winning are still only 1 in 3. That’s the sucker’s choice. Sticking with your guns means sticking with a strategy that will get you a goat 66.67% of the time.
If this is still not clicking, I’d suggest you have a look at the video above. Once you understand why switching is always a good idea, you might be wondering whether this lesson in a defunct game show (Let’s Make a Deal is back on the air, but with an altered format) holds any relevance for the real world. Life rarely presents us with tripartite choices that could yield, in a split second, riches or bupkis. Here is a simulation prepared by Builtvisible for Unibet, an online gaming site, which lets you try your hand at the challenge. It's part of an interactive online article outlining a few ways in which understanding game theory can improve your decision-making. Now, click inside the box, scroll down and press “Start” to get your Let's Make a Deal on.
If you switch as an ironclad rule, you will, with repeated iterations, win two-thirds of the time. If you stay, you will win one-third of the time.
What does this mean? Three things.
First, experts are not always right. I don’t want to encourage radical suspicion of scientists — such irrationality is behind climate-change denial and the theory of “intelligent design.” But the Monty Hall controversy of 1991 was an interesting study in how experts and laymen alike can be completely and embarrassingly wrong. Keep your wits about you when you come across scientific conclusions; even Ph.D.s have lapses of judgment.
Second, your gut instinct is not always right. It pays to question yourself. As I wrote last week, “You Are More Ignorant Than You Think You Are.” A feeling of attachment to your first choice because, as they say, you should “go with your instinct,” bears no marker of rationality.
Third, and related, realize the importance of reassessing a situation when facts change. While there may be no Monty Hall-style choices in your real life, there are plenty of times when new information pours in about an investment, the character of a colleague, or a political issue you’ve been on one side of for as long as you can remember. As John Maynard Keynes said, "When the facts change, I change my mind. What do you do, Sir?"
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