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Monty Hall and all that (and "painfully easy")

There are two groups of people who disagree with the standard solution to the Monty Hall problem: those who know less than the person who came up with the solution and those who know more.

What do those who know more say?

Consider an exactly parallel problem: You buy one lottery ticket and I buy two. I write down your number and my two numbers. I watch the draw and you don't. After the draw, I announce that one of the three tickets has won the million dollar prize and generously offer to trade my two tickets for your one. Should you accept?

It would be difficult to find a 10-year-old child who would accept such a sucker offer, but according to the reasoning of the standard Monty Hall solution this would double your chances of winning.

In a situation where someone with more information than you volunteers some information (as in "painfully easy") or makes you an offer (as in "Monty Hall"), you can't simply use rules of probability based on an assumption of randomness. The question of why this piece of information or this offer was put forward is also significant.

If Monty always shows a goat and offers the chance to switch, you should indeed always switch because it will double your odds of winning. Likewise, if you are convinced that the decision of whether or not to offer a switch is based on factors completely independent of whether or not you have chosen the winning door (or, for that matter, if he only offers the chance to switch in cases where you have lost and he feels sorry for you).

Steve Hutton (stevehutton at rogers dot com)
Wednesday, July 24, 2002

I didn't make this problem up... its a CLASSIC probability problem, and people ALWAYS try to say that the solution is wrong.  You are welcome to your opinion but I must say that the solution is correct, as can be verified by asking anyone who does probability or simply doing a search on the internet for "monty hall problem".

Michael H. Pryor
Thursday, July 25, 2002

So, which part of my reasoning is incorrect?

Steve Hutton
Thursday, July 25, 2002

The Monty Hall problem assumes that Monty Hall doesn't care if you win or lose.  If Monty Hall wants to make you lose, AND he can choose whether or not to show you the goat and offer you a trade, then it's a completely different problem.

So the lottery ticket problem above is not at all like the famous monty hall problem, since your friend wants to win the million dollars, and he doesn't have to offer the trade.

Billy Martin
Thursday, July 25, 2002

Some of the sources qualify the answer (and are thus correct); many don't (and are thus wrong). Here is a good excerpt from the sci.math FAQ:

If the host always opens one of the two other doors, you should switch. Notice that 1/3 of the time you choose the right door (i.e. the one with the car) and switching is wrong, while 2/3 of the time you choose the wrong door and switching gets you the car.

Thus the expected return of switching is 2/3 which improves over your original expected gain of 1/3 .

Even if the hosts switches only part of the time, it pays to switch. Only in the case where we assume a malicious host (i.e. a host who entices you to switch based in the knowledge that you have the right door) would it pay not to switch.

(Out of the FAQ now, and back to me)

So, *if the host always offers the chance to switch*, you should always accept. If the host is deliberately trying to trick you (as in my example), you shouldn't. The unqualified statement "you should always switch" is false.

The only part of this excerpt that I disagree with is the overly broad statement "Even if the host switches only part of the time, it pays to switch." If the host offers you the chance to switch 90% of the time when you guess wrong and 10% of the time when you guess correctly, you shouldn't switch. The FAQ would be correct if it said instead "No matter how often the host offers you the chance to switch, provided that the probability of making an offer is independent of the probability that your original guess was correct or incorrect, you should switch."

Steve Hutton
Thursday, July 25, 2002

Oops. That last note should have said "90 percent of the time when you are *right* and 10 percent of the time when you are *wrong*". The key is that the host is at least twice as likely to let a winner switch as to let a loser switch.

There are three stages that people can go through with this problem:

In stage 1, they don't analyze the probability carefully and conclude that the odds are 1/2 whether or not you switch. Because this is so "obvious", they tend to be vehement.

In stage 2, they have analyzed the probability properly (or, more likely, read an account by someone else who has) and see the error of their ways. They conclude that the odds are 2/3 if you switch and 1/3 if you don't. Because they have just gone through a conversion experience, they tend to be vehement.

In stage 3, they realize (or, more likely, read an account by someone else who has realized) that the standard stage 2 solution is based on a by-no-means-obvious assumption (that the host's actions are completely neutral). They conclude that the question is unanswerable in terms of probability because it depends on a factor that can't be reliably estimated (namely, the rule being used by the host). They may also note that game theory deals with exactly this type of problem, and that under the standard assumptions in game theory (the same assumptions that we would make if we learned that Monty Hall dabbled in piracy), the only reason someone would offer you the chance to switch is that you have already won. Switching would decrease your probability of winning from 1 to 0. Because they have now gone through two conversion experiences, these people tend to be vehement.

Is there a fourth stage I'm not aware of? As a typical stage-3er, I vehemently deny it.

Steve Hutton
Thursday, July 25, 2002

In stage 4, they (vehemently) make sure that whenever they pose the question, they make it absolutely clear that Monty will always open a goat door and will always offer you a chance to switch. Stage 4 people also vehemently correct anyone who mis-states the problem, and vehemently explain why it's important to state that Monty _always_ offers a chance to switch.

Stage 5 people have given up, on the basis that it takes too much out of your life when you try to explain probability to people who know with absolute certainty that an "unknown" probability is always 50%, because either it IS or is ISN'T.

Adrian Gilby
Thursday, July 25, 2002

An excellent analysis and an angle I hadn't considered.  So let's go back to the original statement of the problem (on this website).

you are presented with three doors (door 1, door 2, door 3). one door has a million dollars behind it. the other two have goats behind them. you do not know ahead of time what is behind any of the doors.

monty asks you to choose a door. you pick one of the doors and announce it. monty then counters by showing you one of the doors with a goat behind it and asks you if you would like to keep the door you chose, or switch to the other unknown door.

OK, let's nitpick.

"monty [sic] counters by showing you one of the doors with a goat behind it."

Did Monty know what was behind the door before he showed me the goat or did he just happen to pick the goat?

Is Monty trying to goad me into switching because he knows I picked the right door?

"or switch to the other unknown door."

The door is unknown to whom?  Monty?

William Frantz
Friday, July 26, 2002

"Did Monty know what was behind the door before he showed me the goat or did he just happen to pick the goat?"

I think it becomes apparent that Monty knows what is behind all doors when you think of the problem in its context: as a game show.  He *has* to know where the money is in order to be sure he doesn't reveal the contents of that door.  Otherwise, you would never get the opportunity to switch (or, if you did, it would be pointless) because there is money behind only one door.

Likewise, and as others have pointed out, we must assume that Monty will offer the switch regardless of whether or not your first guess is right or wrong.  Also as a bit of a nitpick, I must suggest that it doesn't matter if Monty wants you to lose (or win, for that matter), provided that he will always offer the switch.  Just like in the "painfully easy" coin problem, the fact that Monty is a human should be disregarded; he's simply a convenient device for the word problem.

Drew Boyles
Saturday, July 27, 2002

If Monty's behaviour is constrained by a rule (for example, always show a goat and offer the chance to switch), his wishes are irrelevant. If his behaviour is unconstrained, and he acts in order to increase the probability of a desired outcome, his wishes matter a lot.

The interesting thing about the "it's bleedingly obvious that Monty always offers a chance to switch and you're just nitpicking if you insist that we spell it out" theory is that in the actual game Monty didn't come anwhere near to always offering the chance to switch.

If you were an actual player on Let's Make A Deal, faced with the actual situation presented in the problem, your best approach would be to analyze Monty's previous behaviour and try to determine the (probably informal) rule that he was following. After performing the analysis, you and your spouse should dress up as a hot dog and a bottle of ketchup in the hopes of attracting Monty's attention and being chosen as contestants.

It was a great show!

Steve Hutton
Saturday, July 27, 2002

Hmm.  I always had problems with this as well until someone described an expansion of the problem which made it clear for me.

Instead of three doors suppose you have 1000.  You choose a door.  Monty opens 998 doors and offers you the chance to switch doors.  When you chose your 1 door you had a 1/1000 chance of being correct.  Monty is offering you the other 999 doors but saving you the effort of opening 998 of them.  I.e., if you switch you now have a 999/1000 chance of getting the prize.

Try it with a deck of cards.  Put 1 red card in a packet of 26 black and have someone play Monty.  You are better off switching.  (Monty knew what was behind the doors on the show and wasn't alwasy on your side.  I've seen interviews where he discusses this.  I'm sure a Google search would dig them up.)

-ljr
Sunday, July 28, 2002

Since Steve Hutton brings the actual Monte Hall into the discussion...

Many years ago (in the 10 to 20 range), I read an interview with Monte Hall about exactly this problem. He said that the producers required that if he showed someone what was behind a door they hadn't chosen, he absolutely was not permitted to offer a switch to the third door.

He could (and frequently did) however, offer them $1000 cash to walk away from the door they chose, or offer them whatever is behind the curtain on the opposite side of the stage, or whatever. Just not the other unchosen door.

BTW, I find Steve's analysis exactly on target.

Jim Lyon
Monday, August 05, 2002

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