Day 166: Let’s Make A Probability Puzzle

originally published June 14, 2012

I may have unearthed the most lengthy and mind-taxing article in Wikipedia about goats. Seriously, this thing reads like a textbook chapter, delving into variants, theorems, and something called the quantum version. It even features this:

I’ll be ignoring that portion.

Today we’re dealing with logic and, to a lesser and more manageable extent, math. For anyone old enough to remember Let’s Make A Deal, you may have heard this one before. If you don’t recall the show, here’s a little refresher.

Let’s Make A Deal was a game show that originated in the early 1960s, and which exemplified more than any other game show on television the you-might-get-screwed-over tenet of capitalism. Contestants were chosen at random by the host – originally Monty Hall, now Wayne Brady – and offered prizes. They were then offered the opportunity to exchange prizes with other audience members, or for the mysterious contents of a box. Sometimes that box contained a better prize, sometimes it contained nothing, or a gag prize. Then the contestant would have to go home knowing they blew a good deal by getting greedy, and have nothing to show for dressing up like this on national television:

The big moment in the show came when a contestant would be offered three doors to choose from. Behind one would be a huge prize, like a car or a trip. Behind the other two, nothing. Here’s where our story picks up.

Imagine you’re standing on stage with Monty Hall (not Wayne Brady – his mathematical paradox probably has something to do with improv and Greg Proops). Monty tells you that behind one of the three doors is a new Kia Picanto.

Behind the other two doors are goats. I’m not certain if you get to keep the goats if you win them, maybe harvest their cheese (that sounds gross), but it doesn’t matter. In theory, you want the car.

You choose Door #1. Monty smiles and you try not to think about how much his jacket resembles your grandmother’s basement couch, the one she still refers to as a ‘chesterfield.’

Monty declares that he will show you what’s behind one of the two doors you didn’t pick. He opens Door #3 and reveals a goat. Now he offers you a choice: do you stick with Door #1, or switch to Door #2?

If you answered with, “It doesn’t matter,” then you really have to ask yourself why there’d be this much Wikispace taken up with the problem, not to mention that scary looking chunk of math up there, if it didn’t matter. Of course it matters. So which do you pick?

Most people’s first guess is that they have a 50/50 chance now that there are two doors left. But that’s not exactly correct. Have a look at the three possibilities:

  1. The Car Is Behind Door #1. If you stay, you win a car. If you switch, you get stuck with a goat, along with the dreary realization that you have a problem trusting your instincts, and will probably suffer the rest of your life because of it.
  2. The Car Is Behind Door #2. If you stay, you get a goat. If you switch, you win the car. Now if you stay, you’ll doubt that your instincts are ever right, and you’ll probably suffer the rest of your life because of it.
  3. The Car Is Behind Door #3. Monty isn’t going to open Door #3 if the car is there, as there’s no drama in watching you choose between a goat and a different goat (unless one is a robot goat, which would be awesome). So he’ll have opened Door #2 instead of Door #3. Now, if you stay, you get a goat. If you switch, you win the car.

As you can see, with two of the three possibilities, switching doors nets you a car. Only if your keen sense of guessery was correct to begin with does switching doors prove to be a mistake.

To put it another way, when you’re offered the original choice, you have a 1 in 3 chance of being correct, and a 2 in 3 chance of being wrong. By eliminating one choice, Monty is essentially offering you the choice between your 1 in 3 chance, or the 2 in 3 chance. Your odds are better to switch.

If your brain is still fighting this solution, imagine the problem featuring a million doors instead of three. First of all, the size of the studio would be incredible. Second, your odds of picking the right door are almost nothing. Now, if Monty opens 999,998 of the 999,999 doors you didn’t pick, then offers you the option to switch to the other remaining door, would you take it? Would you truly believe that you had picked the right door to begin with (with 1 in a million odds), or would you jump at the likelihood that you did not?

There are a few factors that cloud the issue, obviously. First of all, you have to assume Monty knows where the car is. Of course he does, that’s easy. Not only is he the host of the show, but he’s Monty Hall. He knows everything.

Next, in order for this to be a non-muddy problem, Monty would have to behave this way every time. The fact is, he didn’t. Sometimes Monty opened one of the unchosen doors, sometimes he did not. Was he more likely to open one of the goat doors if you picked the car? Only Monty knows that, and he ain’t telling.

This problem has shown up in a number of pop culture venues: Penn Jillette discussed it on Bob Dylan’s radio show, and you can spot it in the film 21, or on an early episode of NUMB3RS. The Mythbusters put it to the test also, and found that not only was there an advantage to switching, but also there was a strong tendency for people to remain hitched to their initial selection.

So maybe the Monty Hall Problem isn’t so much about odds and math and long equations with that funky, zig-zaggy ‘E’ in them, but more about human psychology.

In fact, maybe all of Let’s Make A Deal was about human psychology. Somehow that just makes sense.

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