Dear Dr. Math,
On the show Deal or No Deal, if the contestant gets to the point of only having two cases left they have the option to switch cases. Should they switch or not? Is this the same as the Monty Hall problem?
As Scott Bakula would say, Oh boy. I guess there was no way I was going to get away with writing a math advice blog and not having to explain the Monty Hall Problem at some point. For those of you out there who may be unfamiliar with the MHP, here's the way it goes:
You are presented with three doors and told that behind one door is a car and behind the other two are goats. (Here we're assuming you want the car and not the goats, but in these tough economic times maybe they should be reversed.) You pick a door and then the host, the venerable Monty Hall, always opens one of the other two doors to reveal a goat. He then offers you the chance to switch to the remaining third door. It turns out that it's always in your best interests to switch, given the available information. Doing so improves your chance of winning from to .
Now, I see some of you reaching for that email button, getting ready to fire off an angry letter about how it just can't be true that switching is better than not switching. After all, there are two remaining doors and you don't know which has the car, so aren't your odds 50-50? It's impossible! Believe me, I sympathize, but hold it right there. Plenty of people, even professional mathematicians, have said the same thing as you. Whole books and websites have been devoted to this topic, people have written simulators that you can try out for yourself, the advice columnist Marilyn vos Savant essentially made her career by being right about this problem and explaining why. The MHP is math's version of an optical illusion--you can stare at it and stare at it, but until you actually get the ruler out and measure, you won't be convinced. The sad truth is: Ellen Tigh's a cylon, Darth Vader built C3PO, and switching doors in the Monty Hall Problem improves your chance of winning from to .
Instead of opening up all the old wounds the MHP has inflicted over the years, let me try to offer my own perspective on how I think about the problem (inflicting all-new wounds!), and then maybe we can take those same ideas and apply them to the Deal or No Deal question to show why it's different.
Let's back the train up all the way to the station and talk a little about what probability is--what it means. Warning: Heavy Philosophy-Type Stuff Ahead. As I've mentioned previously, my opinion is that probability is a way to quantify the uncertainty we have about the state of the world. Therefore, it's highly dependent on what information we feel that we possess about the things we observe and what consequences the information may have. For example, everyone's favorite "random" activity is flipping a coin--assuming it's a "fair coin", the probability is that it will come up heads and that it will come up tails. But what does that really mean? Physically, we can model all the variables that go into the action of flipping a coin--weight distribution of the coin, air resistance, amount and location of force applied to the coin, the direction the coin is tossed, elasticity of the landing surface, etc. If somehow we could measure all of these things between the time the coin was tossed and the time it landed, and if we had access to a powerful enough computing device, we could predict whether the coin would come up heads. At the very least, we could guess ("calling it in the air") and improve our chances to more than . Going back a step, the only parts of this system unknown to us ahead of time are the variables due to the tossing itself--the human element of thumb against coin. If, for example, we knew that the person tossing the coin were an amazingly skilled athlete who could control his hand and arm motions with extreme precision and who had practiced the technique of tossing a coin enough that he could reliably make it come up heads, we again could improve upon our 50-50 guess. As a third possibility, consider the case where the coin has already been flipped but we haven't seen the outcome yet (the referee's still holding it); if somebody could sneak a peek at part of the coin and tell us what they saw, we could update our information and make a better guess.
So, what is the "real" probability? In my view, and this might be hard to swallow at first, the answer is there isn't one--the question itself is flawed. "Wait a minute," I can hear you objecting, "Can't we just perform experiments and measure the frequency of heads? Flip a coin a hundred times and about 50 of those will be heads, etc.?" The problem there is that you're observing a different event each time. You can never step in the river twice, nor flip the same coin. All the repetition does is validate the predictive power of your mental model that says that the factors that go into flipping coins are beyond your comprehension and result in the heads side and the tails side being equally likely. As an alternative, say, you could have the mental model (shared by many people) that those hundred coin flips were predestined to occur the way they did and that through meditation/prayer/drugs/etc. you can actually see into the future and predict the outcome of the next flip. It happens that the first model tends to be more successful than the second (or any others) in this instance, but we should be careful to separate the things we're assuming from the things we observe. As E.T. Jaynes wrote in Probability Theory: The Logic of Science, trying to verify the probability of an event by performing experiments "would be like trying to verify a boy's love for his dog by performing experiments on the dog."
See, part of the problem with the way we humans interpret the world is that the physical laws we rely on--for example, that two colliding objects obey the law of conservation of momentum--can quickly outpace our abilities to calculate their consequences--say, the motions of every molecule of a balloon-full of air. We use probability as a way of approximating the behavior of these complex systems instead of having to understand them completely, but that doesn't mean that the events "are" random. A more powerful being might see things differently, the way adults see tic-tac-toe differently from the way little kids do. But we seem to be stuck with this uncertainty about complex systems. And there's really no system on Earth more complex than a human, which brings us back to the MHP.
In the setup to the Monty Hall Problem, we've assumed some things, all of which pertain to the actions of other people. First, there is the assumption that the car is equally likely to be behind any of the three doors (actually, assumption zero is that there even is a car at all). Presumably, some producer or somebody chose which door to put it behind--it's possible they might have had a preference for door #1, for example, because it's closer to the loading dock or looks better on TV. If we had records of thousands of shows, we might gain some insight into their decision process and detect some bias. But we're assuming otherwise. Secondly, and this is the real key, we have the assumption that Monty Hall knows which door has the car behind it. As a consequence, we can deduce that by opening up the remaining door (or one of the two remaining doors, if we initially chose the one with the car), he has added information to the set of things we know about the game. Namely, we know that if the car had been behind one of the other two doors, he would have been forced to open the door he did--that's essentially why switching gives us a chance of winning. If the other door had opened by chance, say a gust of wind blew it open and we happened to see the goat, then we'd have no reason to conclude anything about whether we should switch, because we just as easily could have seen the car. So, by knowing what Monty knows, we can improve our chances. In coin terms, it's as though we had a prearranged deal with the referee where if the coin is tails, he just tells us half the time and stays quiet the other half, and if the coin is heads he always stays quiet--so if he doesn't speak, we know there's a chance the coin is heads.
Now, on Deal or No Deal, hosted by the incomparable Howie Mandel, the situation is somewhat different. For those who haven't seen the show, it works like this: a contestant picks one of 26 briefcases, each containing a different dollar amount. He/she then opens some or all of the remaining briefcases and decides whether to keep going or sell the initial case. In the extreme situation in which he/she keeps going all the way to the end and there are only two cases left, the contestant has the option to keep the original case or switch. Let's you and I pretend that we were on the show. For simplicity, let's assume that initially 25 of the 26 cases had $0, and the one remaining case had $1 million. Also, let's assume that we opened 24 cases and inside each one was a big fat $0 (we got to say "NO DEAL!" a bunch of times, which was fun; also, they brought out Ellen Degeneres at some point). What does that mean about our prospects? Should we switch? Well, our assumptions, again, were (1) all cases were equally likely to contain the million dollars, and (2) nobody on the show knew which case was which. Under those assumptions, it doesn't matter if we switch or not, since the probability is of each case having the million. It's just like the Monty Hall Problem if Monty didn't know which door had the car behind it--nobody has given us any additional information with which to prefer one case over the other. If, however, we knew that Howie knew which case had the winner, and he had started the show by opening all the other cases, then we should absolutely switch in a heartbeat, because it would improve our chance of winning from to . It's all about what information Howie gives us. Also, if he could give us Anya's phone number while he's at it, that would help us out, too.