I’ll agree that to be physically identical you must not show me the step by which the coin got into your hand - I see only your hand slapping the coin onto the table.
But the fact remains that the probabilities strongly depend on your motivation.
I think I see one flaw here. “Monty reveals another door” is not one possibility, but two, since he could pick either of two doors. Therefore there are four outcomes here, not three, and two win.
I am open to an explanation of where I’m wrong.
The four outcomes do not have equal chances. Because the big prize has an equal chance to be between each of the three doors:
You pick number one, Monty reveals another door – 1/3 chance in total, hence 1/6 chance for each of the doors that Monty might coose
You pick number two, Monty reveals three – 1/3 chance
You pick number three, Monty reveals two – 1/3 chance
So the are two outcomes, each with a 1/6 chance, that are losers, and two outcomes, each with a 1/3 chance, that are winners. So if youswitch, you have a 2/3 chance of winning.
As others have said, you need to be careful when calculating conditional probabilities.
Cooking with Gas
Where you go wrong is in modeling the revelation of one door as if it were a random event. If that were the case, you would also have to model the two options in which Monty reveals the dorr you did not pick but which holds the proze. Then you have 6 possibilities: 2 win and 4 lose (but for 2 of them you discover you cannot win before making the choice). Most people omit the 2 “options” in which Monty reveals the prize and asks if you want to switch, because that “never happens” in Let’s Make a Deal. Thus, they reach teh same conclusion you did, that the odds are 50-50, which is true if Monty reveals a door at random AND the revealed door does not hide the prize.
In the actual problem, though, Monty’s choice is not “free”. There is no chance that he will reveal the prize before asking you to choose. Thus, it makes no sense to enumerate the options as if revealing one door might present information to you beyond what you know at the beginning of the game: At least one door that you did not pick does not hide the prize. Monty’s “reveal” does not, in fact, provide you with any more information about the odds of “your” dorr being correct. It does, however, tell you more about the remaining door that you did not pick. In 2 out of 3 plays, that door will hide the prize.
Again, if acting upon a physical objct is done differenctly, then it’s not the same thing.
I’m picturing it this way: Before the show, Monty is told the prize is behind #2. Contestant picks #1. Now Monty happens to be hung over from a night of heavy boozing, and in an instant he realizes he can’t remember what door the prize is supposed to be behind. Flop sweat is moments away; they may have to suspend taping. Thinking quickly, Monty realizes that if he just guesses, he’s got a 50-50 chance at picking a zonk and not having to suspend taping. If he reveals a prize, they can just reset, as they’d have to suspend taping anyway for him to get confirmation as to the prize door. So he guesses, randomly, and picks door #3. Yay! A zonk! Monty breathes a sigh of relief.
(Unlike flipping vs. stacking a coin, this example is actually the same physical setup.)
Strangely, in this scenario, the contestant supposedly has a 50-50 shot regardless if he stays or switches. Simulations be damned, I call bullshit on this. The contestant has double the odds to win if he switches.
Incorrect.
If Monty gets amnesia but the contestant still gets to finish the game, one of two things must happen. Either the contestant chose correctly and so it doesn’t matter which door Monty opens (1/3 likely), or the contestant will have chosen incorrectly but Monty will have inadvertantly revealed the zoink anyway (2/3 x 1/2 = 1/3 likely). Just by getting to this stage we know that Monty didn’t open the wrong door and reveal the prize (2/3 x 1/2 = 1/3 likely)
Monty having amnesia only affects the probability of him opening a door and revealing a zonk. When he doesn’t have amnesia he will do this 100 % of the time. With amnesia he will do it 67% of the time. On the occasions when Monty has amnesia but reveals a zonk anyway, he has rather fortuitously managed not to deviate from the original set up, and so the probabilities for the contestant are unaffected. They remain:
Chance of switching and winning if the contestant:
Chose the correct door initially = 0%
Chose the wrong door initially = 100%
Chances of sticking and winning if the contestant:
Chose the correct door initially = 100%
Chose the wrong door initially = 0%
Chances of initally choosing:
The correct door = 1/3
The incorrect door = 2/3
Overall chance of winning by switching = 2/3
Overall chance of winning by sticking = 1/3
Assuming switch/stick decision arbitrary:
Overall chance of winning = 50%
Overall chance of losing = 50%
The only difference if Monty has amnsesia is that 1/3 of the time the game breaks down and taping has to be stopped. When Monty does not amnesia it is not possible for the game to break down. If taping does not stop there has been no deviation from the standard Monty Hall problem and so probabilities remain the same.
The only way for the probabilities to be affected would be if Monty covers for his amnesia by refusing to open the door he choses, so that rather than risk stopping taping, the contestant is presented 1/3 of the time with a choice between two doors which both contain zonks. This clearly is a radically different set up to the original game, because if Monty without amnesia was allowed to do this he could fix it so contestants who didn’t choose the prize door straight away would always lose. Instead of never revealing the prize he would always remove the prize. In this game, switching would lose 100% and contestants would have a 1/6 chance of winning. 100% of winners would choose the correct door in the first instance.
In the Monty Hall problem, switching wins 67% and contestants have a 1/2 chance of winning. 33% of winners choose the correct door in the first instance.
Here’s the thing: Obviously when faced with two doors to chose between and there really is a prize, the number of winners and losers will be equal. Because ostensively the problem really does reduce to a choice between two doors, people key on the 50% chance of winning and respond that it does not matter whether you stick or switch. What they don’t consider is that clearly only 33% of contestants can be expected to have selected the winning door in the first instance. If everybody always stuck, only 33% of people could possibly win. If everybody switched, 67% of people could win. And as everybody intuitively grasps, if as many people switch as stick, 50% of people win.
When people reject the correct solution it is because the presence of two doors combined with a 50% chance of winning seduces them into instinctively believing the introducion of another fraction would be contradictory. It isn’t.
If 50% of contestants switch and 50% stick then 50% of contestants win. Everybody agrees with this intuitively.
But because
33.3% of contestants choose the winning door first
66.6% of contestants choose the losing doors first
and because 50% switch and 50% stick, that means
16.6% of contestants choose the winning door first and stick and win
33.3% of contestants choose the losing doors first and switch and win
33.3% of contestants choose the losing door first and stick and lose
16.6% of contestants choose the winning door first and switch and lose
So that:
16.6% of contestants are winners who stick
33.3% of contestants are winners who switch
33.3% of contestants are losers who stick
16.6% of contestants are losers who switch.
and overall:
50% of contestants are winners
50% of contestants are losers
50% of contestants switch
50% of contestants stick
But
66.6% of winners switch
33.3% of winners stick
33.3% of losers switch
66.6% of losers stick
33.3% of winners choose winning door first time
66.6% of winners choose losing door first time
66.6% of losers chose losing door first time
33.3% of loses chose winning door first time
Most winners choose the losing door first because most contestants choose the losing door first in a game that lets half it’s contestants win. This is why switching improves the odds of winning. If equal numbers of contestants won by sticking as they did by switching, then the odds of winning would be the same as the odds of picking the winning door straight away, 33.6% of contestants would win and 66.6 % of contestants would lose.
After the contestant makes his initial selection, when Monty offers the chance to switch and doesn’t have amnesia, he fixes the game so that the contestant has either a 100% chance of winning or a 0% chance of winning, depending on whether he has already selected the winning door. Already having selected the winning door is half as likely as having selected a losing door. Therefore the number of contestants with 100% winning odds if they switch is double the number of contestants who will have 0% winning odds if they switch. As contestants don’t know which camp their initial choice has left them in, they should realise they are twice as likely to pick the wrong door first than the right door, and so the better gamble is always to switch.
- 33.3% pick right door first
- 50% contestants win
- 66.6% winners swich.
There is no contradiction.
Monty’s eating salad in the network canteen with hunched shoulders and a nervous disposition. He’s been a professional game show host for a long time but lately he’s been getting these hot flushes under the studio lights and he’s paranoid his boss is going to fire him. His trophy wife thinks it could be the onset of alzeimers disease. Across the room today’s contestants are staring at him as though he were barely recognisable as the king of light entertainment television.
Suddenly an ominous looking man wearing a stern look and a dark suit bursts into the canteen screaming “You’re fired!”. It’s the network’s health and safety executive. Monty closes his eyes as he trys to dissolve into thin air and not think about divorce attorneys.
But the man storms straight past Monty. Instead he begins a stand-up row with the canteen caterer, dismissing him on the spot. Monty overhears everything. The chef’s just imported a bulk quantity of genetically modified avocados from a farm in Bogota that the Feds are investigating for dope smuggling and money laundering. Toxicology reports indicate there’s something dodgy in his guacamole that could induce amnesia in pigs.
As Monty wipes his brow and sighs with relief the tension drops visibly from his shoulders. But for only a moment. Now two more network suits are in his face barking ominously. They’re younger than Monty but they’ve both seen the movie “Quiz Show” and they both know Federal fraud investigations into fixing TV game shows are very bad news indeed. Apparently some young lawyer in the district attorney’s office is following up claims by recent Monty Hall show contestants that they were cheated out of their rightful winnings. They’ve reported woozyness whilst filming, confusion over decisions taken only moments earlier. The DA is planning to prosecute the network. There’s a rumour the young lawyer can prove Monty’s executive producer deliberately told the chef to poison contestants in order to afflict their memory and reduce the show’s budgetry expenditure on prizes. The chef only started work a week ago and it is not clear whether the small number of complainants actually ate the guacamole, but they all claim they did. At no point did Monty actually have to stop taping, although at times he feared he would.
The men tell Monty that because he is a much loved celebrity who the public trust, it will be left to him on the witness stand to persuade the twelve jurors that nothing untoward occurred.
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Is Monty being asked to commit perjury?
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Would the executive producer’s alleged plan have succeeded if implemented undetected?
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Does mutual amnesia give more disadvantage to the contestant or the host?
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Does the fact the show never had to stop taping prove a) Monty had a lucky streak b) the executive producer had a safety net c) the contestants are lying d) nothing
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What would a statistical review of affected shows compared to unaffected shows reveal?
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Would that review exonerate or incriminate the executive producer? Or the contestants?
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If the defence, the prosecution, or both, didn’t understand the Monty Hall problem, who would it hinder, who would it benefit?
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If the jury didn’t understand the Monty Hall problem, who would it hinder, who would it benefit?
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Is conveying the correct understanding of the Monty Hall problem to the jury beneficial or detrimental to the prosecution’s case? In order to win, how should they present it to the jury?
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Is conveying the correct understanding of the Monty Hall problem to the jury beneficial or detrimental to the defence’s case? In order to win, how should they present it to the jury?
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What verdict should the jury reach?
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What crimes have been committed?
Although the two scenarios — (1) Monty revealing a zonk only by chance, versus (2) Monty guaranteeing he reveals a zonk — look physically the same to the player, they really are different. It helps to remember that a probability is a statistical quantity. It’s the approached ratio of desired outcomes to total outcomes as the number of trials increases indefinitely.
(Actually the outcomes of interest might not be so “desired”, if we’re talking about cancer risks or something. But you know what I mean.)
So the two different variants of the game would clearly show their difference if you played each one a hundred times in a row — or alternatively, had hundreds of players playing the games separately, in parallel.