Be sure you read The Monty Hall Problem before you read this node...

Monty Hall himself, presented the problem with the original Monty Hall Problem. The original problem, as proposed by Marilyn vos Savant is mathematically sound, but there is a loophole.

Monty Hall presented a series of trials exploiting this loophole.

Note: The following text originally appeared in the New York Times, January 21, 1991.

On the first (trial), the contestant picked Door 1.

"That's too bad," Mr. Hall said, opening Door 1. "You've won a goat."

"But you didn't open another door yet or give me a chance to switch."

"Where does it say I have to let you switch every time? I'm the master of the show. Here, try it again."

On the second trial, the contestant again picked Door 1. Mr. Hall opened Door 3, revealing a goat. The contestant was about to switch to Door 2 when Mr. Hall pulled out a roll of bills.

"You're sure you want Door No. 2?" he asked. "Before I show you what's behind that door, I will give you \$3,000 in cash not to switch to it."

"I'll switch to it."

"Three thousand dollars," Mr. Hall repeated, shifting into his famous cadence. "Cash. Cash money. It could be a car, but it could be a goat. Four thousand."

"I'll try the door." "Forty-five hundred. Forty-seven. Forty-eight. My last offer: Five thousand dollars."

"Let's open the door."

"You just ended up with a goat," he said, opening the door.

Mr. Hall continued: "Now do you see what happened there? The higher I got, the more you thought the car was behind Door 2. I wanted to con you into switching there, because I knew the car was behind 1. That's the kind of thing I can do when I'm in control of the game. You may think you have probability going for you when you follow the answer in her column, but there's the pyschological factor to consider."

He proceeded to prove his case by winning the next eight rounds. Whenever the contestant began with the wrong door, Mr. Hall promptly opened it and awarded the goat; whenever the contestant started out with the right door, Mr. Hall allowed him to switch doors and get another goat. The only way to win a car would have been to disregard Ms. vos Savant's advice and stick with the original door, defying the Monty Hall Problem Solution.

Was Mr. Hall cheating? Not according to the rules of the show, because he did have the option of not offering the switch, and he usually did not offer it.

Why do people find statistical puzzles such as the Monty Hall problem so difficult to answer? Why do they so often supply 'the wrong answer'? Why do they often stick to their solution even when 'the correct solution' is pointed out to them?

I think I know. The reason is that all of these problems have unstated assumptions, assumptions that are required to model the problem with statistics. Often, the provided answer starts and ends with the mathematics, but that is completely beside the point: the question is how you arrive at the model on which the computation is based; often, the difficult part is what the unstated assumptions are.

In the Monty Hall problem, the unstated assumption is: Monty Hall knows which door has the prize, and always opens an empty door. This is what allows us to view the situation as one out of a repeatable series. And that is an absolute requirement for applying statistics in the first place: you have to know the set of potential events the particular event is drawn from in order to tell its probability, because that is what statistical probability is, by definition.

In the case of Monty Hall, the unstated assumption is fairly obvious, but in some related problems (see the rec.puzzles FAQ for a list) there are several reasonable, conflicting assumptions: they lead to different outcomes. A good example, in my opinion, is the puzzle in a family of two children, of which one is a boy, what are the odds the other is a boy as well?

Often, it is simply unclear what can be assumed, in which case statistics cannot be applied, either. Example: what is the probability that my age is over 20? You can't apply statistics to this question without assuming a population from which I'm drawn: all humans? all Everything noders at the time you read this? All Everything authors? The question doesn't imply any, so the only correct answer is: sorry, but you'll have to tell me before I can give you a statistical probability. A stronger example: what are the odds of intelligent life existing elsewhere in the universe? Here, we cannot even begin to answer without stating some fundamental assumptions on which factors have been critical in the emergence (creation if you wish) of life on earth, and how often they recur elsewhere; and let's be honest, nobody knows.

Therefore, I consider all answers to such problems that consist merely of statistical formulas stupid and more wrong than an 'incorrect' numerical outcome could possibly be.

PS to Geez: in daily life, when we talk about chance, we very rarely have a neatly defined solution space in the way that statistics assumes, maybe it takes a while to get used to the concept.

I think rp has hit the nail on the head here. (When I first read this, I had to glance back up at the top to make sure it wasn't one of my old nodes.)

Someone who does not make the assumptions that Monty Hall knows where the car is and that he will always open a door containing may instead assume that Monty always open a door entirely at random, and if that happens to be the one with the car, then you just lose immediately. In that case, there is no advantage to switching; 1/3 of the time you will have picked the right door, 1/3 of the time you will have picked the wrong door and Monty will open the other door which does not have the car, and 1/3 of the time Monty will reveal the car and you will not have any opportunity to switch doors. Furthermore, after Monty opens the door, in this case, the cases where Monty shows the car are now known to have not occurred, and the others are equally likely -- no advantage to switching.

This is how many people reach the "incorrect" answer that there is no advantage to switching, when the information that is supposed to be given about Monty's behavior is omitted from the problem. This isn't always an unstated assumption, and I seem to recall that in the problem's first appearance on rec.puzzles way back when, this information was included in the problem statement, but it is exactly the sort of thing that gets omitted often when this puzzle is retold.

The other problem with the Monty Hall problem, and this assumption in particular, is that the actual show Let's Make a Deal did not work like this. This was a game with no consistent rules, and Monty quite often, seemingly intentionally, did exactly the opposite of what you might have expected him to do or what he had done before in a similar situation.

When considering the Monty Hall problem, you must first discard the actual behavior of Monty Hall on Let's Make a Deal. The problem is stated clearly, and excludes any distractions or alternate monetary offers. With the two sons problem, we should make the common-sense assumption that the odds of bearing a boy is 50%. Stick to the problem at hand, and not a related problem.

Over-thinking things tends to be the cause of many problems. The Monty Hall problem does not have a deceptive host, it has an automaton doing what is explained. Those sons aren't arranging themselves in clever families to confuse the issue.

The other problem is confusion between chance and reality. People expect a coin to come up heads exactly half the time it's flipped. They include cases that could not happen in the terms of the problem, like including families with 2 girls in the odds of having two sons. Or, they discard information from memory that still applies, like believing that each remaining door has a 50% chance of having the car.

The final problem is that those who have solved the riddle forget that it's supposed to be puzzling. Not everyone is as smart as you. Not everyone has a logical mind. If everyone could think things through perfectly, then many interesting games would be boring. Or perhaps we'd just over-think everything, and we'd all be as annoying as Vizzini.

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