In the Monty Hall Problem a contestant is given a choice between one of three doors, with a fabulous prize behind only one door. After the initial door is selected the host, Monty Hall, opens one of the other doors that does not reveal a prize. Then the contestant is given the option to switch his or her choice to the remaining door, or stick with the original selection. The question is whether it is better to stick or switch.

The answer is that it is better to switch because the probability of winning after switching is two out of three, whereas sticking with the original selection leaves the contestant with the original winning probability of one out of three. Why?

The trick to understanding why this occurs is to view the situation not from the contestant’s viewpoint, but from Monty Hall’s. At the outset, from Monty’s point of view, the contestant has a one out of three chance of guessing the correct door. In the likely situation (two out of three) that the contestant chose wrongly, Monty then has to know where the prize is among the two remaining doors in order to open a door that does not reveal the prize. So Monty opens a door not revealing the prize and asks the contestant whether he or she would like to switch or not.

However, the contestant knows that in the likely (two out of three) situation that the initial choice was wrong, Monty had to know where the prize was in order to open the door that did not contain the prize. Since the contestant knows that Monty has to know where the prize is to make the correct choice, the contestant can (in this likely case) place him or herself in Monty’s shoes. At this point Monty knows that the remaining door is the one that contains the prize, and hence the contestant should switch.

If we consider the unlikely situation in which the contestant initially chose the door with the prize behind it, then this line of reasoning will not work. Imagine that Monty forgets the location of the prize every time the contestant guesses correctly. In this situation he can still open either of the remaining doors without ever ruining the game. From his perspective the location of the prize is unrelated to his actions; it played no part in his decision to open one door or another (he merely chose a door the contestant hadn’t).

So, in the one out of three case where the contestant initially selected the correct door, there is no way to deduce whether switching is beneficial based upon placing oneself in Monty’s shoes: the situation where Monty has forgotten the prize’s location is indistinguishable from a situation in which he has not forgotten. Without any way to further analyze the situation and tilt the odds to over one out of three, the contestant should always assume that he or she is in the previous, more likely, situation and take the opportunity to switch.^{1}

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^{1}Imagine that the contestant has a guardian angel that will let the game run its course if the contestant switches doors, but will change the location of the prize such that if the contestant sticks with the original door the angel will make sure that the contestant wins four out of five times. Then the probability of winning while switching will stay at 2/3 but the probability of winning while sticking will be 4/5. If the contestant had some way of divining that this was happening, this would be a case in which further analysis would be of benefit.

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On 13 Aug 2009, 13:48.

This is a game, and it is likely that in such a game the house (Monty) does not want you to win. This is the ordinary case. If such is the case, then obviously the house knows where the prize is. If the house does not want you to win and the house knows where the prize and the game is not always the same…meaning you might just have Monty open your box and say, “You lose,” then you might be wiser not to switch as it is likely that Monty is trying to get you to switch. If the game is a one shot deal and he can open your first choice and say, “You lose,” why would he give you a choice?

Regards,

Timothy E. Kennelly

Those are the rules of the game. Check out the Wilkipedia entry for a bit of background.

If one works with the rule that the house always gets rid of one choice and asks if you want to switch, then you should switch. My point is that that rule is not necessary and generally one should only assume that the house wants you to lose – that being the case, you really should not switch.

On “Lets Make a Deal” there was no rule that Monty would get rid of one choice and asks you if you wanted to switch, and likewise if you played blind three-card-Monty on the street (if there is such a thing) then there is no such rule and switching would generally be foolish.

Regards,

Timothy E. Kennelly

Hi Tim,

There isn’t actually any question whether switching is better or not. I suggest you play one one of the many Monty Hall games online. Also check other people’s sites. Don’t take my word for it, but switching gives you a 2/3 chance of winning. The only question is why.

The toy game is constructed so that switching always gives one the one choice in two advantage over the one choice in three, my point is that no game you would actually play in the real world is like the toy game. In the real world, when Monty shows you 3 boxes, he doesn’t say, “Now you pick one, and I will cut out one that is not the winning box, then you can switch if you like.” Such a game is not a normal game. In a normal game, Monty does what he likes, you pick and he might reveal that you lost or cut one and give you a choice and he know where the prize is and he generally does not want you to win. Although, Monty is also an entertainer, so he would likely let the contestant switch sometimes when switching results in a win.

In any case, the toy game is too far from reality, it does not reflect the real likely interests of Monty or the house.

Regards,

Timothy E. Kennelly

Hi,

Sure- this game is unlikely to ever be played again in real life because of the probability favors the player too strongly. However, it is philosophically interesting because of

whythe odds are so skewed. In this sense, it is only a philosophical and mathematical topic.I appreciate your sentiment that no house should ever play this game; it indicates that you understand that the probabilities are unusual in this instance. However, not everyone agrees with me or even believes that the odds are as they really are. There are hundreds, if not thousands, of places online where you can find someone discussing this problem.

What you see above is just my two cents on why the odds are the way they are, and don’t have anything to do with a current real word situation.

Hi, (Indeed!)

Your comments about the probability of winning and losing are undoubtedly correct. If the rules of the game are really that Monty always give you three choices from which you pick one, followed by Monty getting rid of one and you having a choice to change your pick – yes, one should always change to the other box.

Regards,

Timothy E. Kennelly

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