Scientific American math and physics editor Davide Castelvecchi revisits the Monty Hall problem, so you can know whether you're better off holding on to your original pick or switching when new information presents itself. Web sites related to this episode include Revisiting the Monty Hall Problem
Steve: Welcome to the Scientific American podcast, Science Talk, posted on June 25th, 2011. I am Steve Mirsky. This week on the podcast:
Castelvecchi: So Monty never picks door number one, but you know that he'll never pick the door that has the car.
Steve: That's Davide Castelvecchi. He's one of our editors at Scientific American magazine, specializing in math and physics and he's talking about the infamous Monty Hall problem I've been promising you this episode for awhile. Davide wrote about the old Let's Make a Deal conundrum in a recent blog post, so we decided to delve again into the decades old poser of whether you increase your chances of winning if you change your original pick from door number one to door number three after Monty opens door number two. And away we go.
Steve: Davide, what's behind door number one?
Castelvecchi: We don't know. That's what we're here to find.
Steve: Why are we examining the Monty Hall problem again, which was in the news like 20 years ago? So how did it come up again for you?
Castelvecchi: One good reason is that it's fun, it's fun to think about, but the specific reason was that we had a column in our January issue, and it was written by a statistics professor who is also a writer of popular books, called John Allan Paulos. And the column was about how animals are or are not skilled at understanding probabilities, and he was making the point that sometimes animals may be better than humans at guessing probabilities, and he made the example of the Monty Hall paradox or problem.
Steve: Because some pigeons were actually able to learn fairly quickly that you should—well, actually we should go through the Monty Hall problem, I guess, for people who don't know it—but the pigeons got it within a few moves and some people just refused to get it, ever.
Castelvecchi: So, the pigeons, sort of, experimentally found their way to the correct answer, whereas people who, kind of, overthink it, get it wrong.
Steve: So, what is the Monty Hall problem? Maybe people don't even remember, Monty Hall used to be the host of Let’s Make a Deal, which I think is back on TV today.
Castelvecchi: So, the game was actually or the problem was not invented by him, but it was inspired by his game, and it was called the Monty Hall problem by a statistician. And it became famous, or notorious, we should say, in 1990 when a writer called Marilyn vos Savant wrote about it in the magazine Parade.
Steve: Which is read by millions and millions of people.
Castelvecchi: Correct. And she explained the strategy of this game, and she got something like 10,000 letters from angry readers who thought she had gotten it wrong. So what's the game? The game is, there are three doors, and if you're the player, you are trying to win a car, and you know that behind one of the doors hides a car; you don't know which one. The other two doors hide two goats, one goat each, and so you take your pick. You pick a door and next Monty picks another one of the remaining doors and opens it, and reveals a goat.
Steve: Right, he always picks a door that you didn't pick, but that definitely has a goat behind it to show you.
Castelvecchi: Correct. So at that point, he gives you a choice. He says, "Do you want to stick to your original choice or do you want to switch to the remaining closed door?"
Steve: Okay, so now the usual reaction to this problem is, I had a one in three chance. Now I know that one of those two doors that I did not pick was not the right door, so I might as well stick with my original choice because it was, at worst, a one in three chance and depending on how I look at it, maybe it's a one in two chance right now. So I might as well just stay. I don't gain anything by switching to the other door, because that's also going to be a one in three or one in two.
Castelvecchi: Right. And that's our, like, anyone's first, our intuitive choice would be that probably. But interestingly even people who are very well versed at statistics, even statistics professors, do calculate, they take paper and pencil, and they calculate. And they come up with the strategy that it's better to stick to your earlier, you know, it doesn't matter; that if you stick to your original choice, you have an equal chance of winning than if you switch.
Steve: Whereas the real answer is, you have a much better chance of winning if you switch.
Castelvecchi: It's always better to switch.
Castelvecchi: And here's why. Let's say that the door you picked originally was number one.
Steve: Okay, I picked door number one on the far left.
Castelvecchi: Now there are three possibilities in this case. I mean, there are three possibilities always—that the car is behind number one, number two or number three. Now, if the car is behind door number one, the door you picked, then Monty has a choice whether to open number two or number three. So let's say he picks randomly one of the two.
Steve: Right, it doesn't matter, they're the same.
Castelvecchi: If the door you picked was hiding a goat, then Monty has no choice. He has to open the other door that's hiding the goat.
Steve: Right. So if a goat was behind door one, instead of the car, then only one of the two remaining doors, two and three, has the goat. Monty's revelation of the goat has been forced.
Castelvecchi: Right, but what's really confusing people is that they think in terms of conditional probability. So that's when you have for example a stack of cards. Let's say you have 52 cards and you pull out an ace, and then you ask, "What are the odds that the next card you pull is also an ace?" And that's what is called conditional probability: What are the odds that A is true given that you know B?
Castelvecchi: In this case, however, this doesn't apply. It's not really the right way to think of it, because what are the odds that the car is behind door one, given that Monty opened door two, is not really interesting because he will never open door one, so the fact that he opened door two gives you absolutely no information about the odds that you have picked that the car is behind door one.
Steve: Right, because we're dealing with an agent, in the Monty Hall issue, who has knowledge about the identities of the things behind the doors. In the card situation, it's still a completely probabilistic kind of a situation. The odds of picking an ace the first time were four out of 52. If the first one you picked was an ace, the odds on picking an ace the second time, assuming a completely randomly shuffled deck, are three out of 51.
Steve: But with the doors in the Monty Hall problem, it's not necessarily the case because somebody has intelligently designed the problem.
Castelvecchi: So people, I think people get confused because they think of it in terms of conditional probabilities, whereas they should just think of it in very simple terms. The odds that the car was behind number one are one third at the beginning, and the information of which door Monty opens next doesn't affect that, doesn't change that. And so they're still one third, even after Monty opens the door. And because that's still one third, and there's only one other possibility now, the other one must have a probability of two thirds.
Steve: Right. It can't be, you might think it's a half because once he shows you door number two and there's the goat, then you might think "Well now it's either behind door number one or door number three, so the odds have been increased to a half." But you still might think, "I might as well keep my original pick because it's equivalent to door three." One in three are both one-half likely to have the car. But that's not true, because we know that Monty knows where the car is. Therefore, my initial pick was one third, knowing that Monty knew. The door he reveals is taken out, that leaves two thirds of a probability left in this entire range of it, which all have to go to door number three.
Castelvecchi: Correct. So Monty never picks door number one, which means he doesn't give you any information about what's behind door number one, but you know that he’ll never pick the door that has the car. So, the door he picks gives you information about the other door he could have picked, and so it tells you that the other door has a higher probability.
Steve: So, the reason that this is so confusing to people is because it seems like a straightforward probability question, where there are three choices; whereas in fact, in the middle of this entire sequence of events, there's been a change in the rules of the game.
Steve: Yeah, the intercession of Monty Hall in between the rounds of picking actually changes some of the conditions of the problem.
Castelvecchi: Now, there are also variations. If you give Monty the choice, if it's okay for him to pick door number one, the one you've picked and open that one, or if he has the choice not to pick anything and to just let you decide, then the game becomes much more complicated and there is a lot of, you know, you have to try to guess what he has in mind, it becomes more complicated.
Steve: I would have thought it would become probabilistically simpler if he doesn't show you anything. It's one third or it isn't; each door is a third in that case.
Castelvecchi: But he can decide to make it easier for you by opening the door you opened; for example, if it has a car, or he can decide to make it harder for you.
Steve: I see. So his decision in that case then also becomes part of the conditions of the problem.
Steve: His condition of whether to show you anything.
Steve: Now another confounding issue here is, changing your pick only increases the likelihood of you being correct. It doesn't ensure that you're going to be correct.
Castelvecchi: It doesn't.
Steve: Right. Your odds are still what they are. So you could change your pick to a much more, two thirds is twice the chance of one third. You could change it to the two thirds and still be wrong and then people would think you're a fool.
Castelvecchi: No, but the idea is, what is the strategy, that if you picked a strategy before you even start playing, what's the strategy that more often, that's most often likely to make you win; so if you play the game, you know, a million times, one third of the time you lose, but two thirds of the time you win.
Steve: Right. The problem is you usually only get to play this game once. So you have to just go in with a strategy that maximizes your chances, and then hope for the best.
Steve: So, this nevertheless continues to drive even professional mathematicians crazy.
Castelvecchi: It always puzzles people and people always argue about it. We've had lot of, even after I posted in my blog, there's been a long discussion in the comments section, and it's a passionate discussion between people who don't buy it, don't buy the solution, and people who try to explain the solution to them.
Steve: So, people can go to the Web site, read that article and chew on this at their leisure.
Castelvecchi: And make up their own minds.
Steve: Either correctly or incorrectly. You can find Davide's blog on his Web site, which to show you how smart he must be is simply sciencewriter.org; and the Monty Hall post is in the Recent Posts section. It's titled "Revisiting the Monty Hall Problem." And speaking of visiting, I'm off to the Lindau Nobel Laureate meeting in beautiful Lindau, Germany. This year's meeting concentrates on physiology and medicine. I already have interviews scheduled with a few of the two dozen Nobel winners, including recent awardee Ada Yonath and the legendary Christian de Duve. I hope to be posting those conversations very soon. In the meantime, get your science news at our Web site, www.ScientificAmerican.com, where, you can check out our blog, called Observations, written by members of the SciAm staff, and by our team of outside bloggers. And remember, if you follow us on Twitter, you'll get a tweet each time a new article hits the Web site. Our Twitter name is @SciAm. S-C-I-A-M. For Science Talk, I'm Steve Mirsky. Thanks for clicking on us.