Pesky five minute edit limit. I wanted to add: “We have some nice consolation prizes for you back stage, including a year’s supply of Rice-a-Roni. Rice-a-Roni, the San Francisco treat!”
More seriously, I think the explanation “would you choose 99 doors, or one” is still confusing (although valid as far as it goes). I would frame it slightly differently.
Let’s keep in mind two basic ground rules here (which are indeed not necessarily true about the real game show):
(1) The good prize was randomly, and secretly, placed behind one particular door before you (the contestant) made your first choice, and will be left behind that door throughout the game. In other words, the people running the show are not going to shuffle things around between the time you make your first choice and when you decide whether to switch. This is a crucial point.
(2) “Monty” knows which door contains the good prize. He will always, after you initially make your choice, then open a door that contains a “zonk” (a bad prize), leaving the door you chose closed, as well as one other door you didn’t choose. If you did not initially choose the good prize, he will never open the door to the good prize and show that you missed out on it (that would take away the whole point of giving you another choice, right?).
So, with those rules understood, let’s say that contestants don’t understand the math behind the “always switch” rule but have had enough experts (who have learned their lesson) insist it’s the right play, that they go ahead and do it (figuring that even the way they understand it, it’s not *worse *to switch, so what the heck). When do they regret switching? Only when they chose the correct door to begin with. That’s it: that’s the only time switching is bad, because you had the right door but you switched it.
So what’s the chance the contestant picked the right door to begin with? 1/3, roughly 33 percent. What’s the chance the contestant picked a “zonk” to begin with? 2/3, roughly 67 percent.
So, again: the “always switch” contestants only go home unhappy the 1/3 of the time that they happened to select the right door to begin with. The “always keep my original pick” contestants go home unhappy the 2/3 of the time they didn’t select the right door to begin with.
I wonder, btw, if the root of some of the trouble here lies with a concept I hadn’t been aware that any intelligent person held until I read the book Sex, Drugs, and Cocoa Puffs by Chuck Klosterman. It is one of my very favourite books, and most of it is absolutely brilliant. But Klosterman occasionally advances notions that I find surprisingly wrongheaded, and the worst of them IMO comes when he writes the following (p. 148):
I’ve always held out some hope that he was being drily sarcastic; but in context and knowing the way he writes, I sadly think not (and he even messed up his probability notation: 2:1 is more like 67/33 than 50/50).
I brought this up to some people I used to play poker with, and was taken back to find that almost all of them agreed as well! “So”, I asked in disbelief, “if you have a gutshot straight draw with only the river card to come, you think it’s a 50/50 shot as to whether you’ll hit your straight?” When they agreed that it was, I shut up and resolved to play as much poker against these folks as possible, LOL…
How did you gather that? The fact that any one choice in this situation is going to be right or wrong has nothing to do with the* chances *of any one choice being right or wrong. It’s an irrelevant point, and incorrectly expressed as a 50-50 probability.
I don’t need to explain this to you, but for the others who insist that the chances of winning are 50-50 and switching has no advantage:
Go buy a lottery ticket. When the drawing is held, don’t look at the winning numbers. I’ll tell you all of the numbers that you didn’t pick and didn’t win. Except for one if you have the winning numbers. So now that you have your ticket, and I’ve pointed out the numbers that A) won, or B) are random non-winning numbers. Would you switch to the ticket with those other numbers if you could? Remember, I can’t tell you which one is winning number, but I can tell you every one that isn’t a winner, except for one if you happen to win.
Ok, that was two posts back. Yes, I didn’t pick up that it wasn’t the Monty Hall. I don’t see how that puts me in agreement with Klosterman. I’ll say it as best I can, the fact that you have two possible outcomes describes a binary choice, not the chances of the outcome of that choice.
I play Hold’em regularly with a guy who believes that eights occur more frequently than sevens or nines which occur more frequently than sixes or tens and so on down to Aces and Deuces being the most rare. (I suspect that he read a book on craps once and thought he could somehow apply dice odds to cards. :rolleyes:) Like you, I keep my mouth shut and always remember to praise him for his ‘courage’ when he manages to suck out against the odds.
As for the Monty Hall problem, the fact that Monty knows where the prize is and will only reveal losing doors is what separates it from Deal or No Deal, where the contestant opens the doors/briefcases, i.e. at random.
TriPolar, I thought your statement “Any one decision like this is either right or wrong” was ambiguous and *might *mean you agreed with Klosterman’s “it’s all 50/50” philosophy. I did not intend to come across as stating with any certainty that this is what you meant, and was honestly just asking you to clarify whether that was what you meant. I got my answer (no, you did not mean that), and I apologise for inadvertently insulting you!
Think of the probability of winning as describing the payoff you’d need on a $1 bet to make it a fair game. In that case, it does apply to a single game.
By a fair game I assume you mean something like flipping a fair coin. If that is the case ‘fair game’ describes the chances of either side appearing on any one flip, but the simple fact a coin provides a binary choice does not. You have to describe how the coin will be manipulated, not just the fact that it has two states.
Which, in case it’s not clear, is not the same as 50/50. A 1-in-100 payoff is a fair game, if the game costs $1 and the payoff is $100 when you do win.
I’m less convinced. As it happens, Sex, Drugs, etc. is one of my current bathroom books*, and I read that line a few days back. The problem is that S,D&CP has a lot more of those goofy, quasi-epigrammatic blobs of text than books like IV or Eating the Dinosaur, and they’re hard to take seriously. In this case, mentions of authority figures and enslavement made it feel a little tongue-in-cheek to me (and I have to say, not all of Klosterman’s blurbs seem intended to make a point, so it could just be him throwing off a joke).
Your mileage may vary.
you either know what that is, or ya don’t…but I’m guessing in the Cecil community you more likely do
Au contraire, I thought this was exactly the thing that was up Mythbusters’ alley. They took a common misconception that has high pop culture awareness and put it to a clear physical demonstration to find the truth. Actually, they added the bit about whether people will change or not, but then looked at the essence of “The Monty Hall Problem” with a clear explanation of the problem, including the necessary caveats, and then a physical demonstration that visually displayed the results of “switch” vs “stay”.
Interesting approach, to cast the person in the role of “Monty” instead of the contestant, but constraining “Monty”'s actions. FWIW I had exactly a 10w 5 l result using your data set and the strategy of switching every time.
Not correct. You are only looking at a subset of the full data. If the host knows where the prize is, his actions are constrained. You have a 1/3 chance of initially picking right and a 2/3 chance of initially picking wrong. When Monty reveals a door, he does not change that probability because he actively selects which door to reveal. When Monty has to guess, there is 1/3 possibility he will pick the winning door to reveal. That changes the total odds. For your described case, you are taking 1/3 of the cases out of contention. The remaining doors are then equal chance.
Yes, for that person in that situation, the odds of having guessed the correct door are 50:50.
Yes, for Joe, there is a 33% chance he chose right and a 66% chance he chose wrong.
Yes, the odds are simultaneously different for two different people who faced two different circumstances, i.e. two different paths to their choices. You and Joe can now be side by side, looking at the same two doors, having coincidentally chosen the same door, but without knowing the circumstances of each other, each of you has your own set of probability for whether you are right or wrong.
There is nothing magical about it. You were never given the choice of one of the doors. That 1/3 probability from Joe is eliminated from your set of choices, so you face a 1/3 probability left is the prize, 1/3 probability right is the prize, and 1/9 probability that the door that was removed would have been on the left, a 1/9 probability that door would have been on the right, and a 1/9 probability that door would have been in the middle. That 3/9 of probability was removed from the problem, evenly from your choice of left or right. Thus you now face a 50:50 condition.
In other words, the door that was removed could have been the leftmost door, and so your choice of “left” would have been that door if it were still present. Or it could have been the rightmost door, and your choice of “right” would have applied. Or it could have been the middle door, and then either the “left” or the "right’ door from Joe’s perspective was removed, e.g. when you enter the room, what you are calling “right” was in Joe’s middle. Because the odds are even it could have been the left or right door, it balances out the missing 1/3 probability that Joe had that you don’t.
Because at any time you could have picked the Million Dollar case had had it taken out of contention. That is the difference - in the classic problem, Monty will never reveal the prize early. You can.
The whole point of the problem set up is to show that the knowledge and actions of the host affect the probabilities. A host who must reveal a door and offer the option to switch, and cannot reveal the prize, is a special case of not changing the starting probabilities, but lumping two options as a single choice. A randomly or maliciously or otherwise acting host can change things significantly. To ignore the role of the host is to ignore the point. Of course people know 2 [del]heads[/del] doors are better than one.
Not every description. Certainly some do, because the person stating the problem had only heard it or doesn’t understand the constraints, or just assumes them and forgets to tell the audience, or whatever. But there are situations where they are explicitly stated. Like that Mythbusters episode, for instance.
Just because the choice isn’t the final choice doesn’t make it less of a choice. And changing the arrow from “I get what’s behind this door” to “I get what’s beind this other door” is certainly a switch.
Correct, mathematically the initial selection does not have to be done by the contestant. You could reword the problem where the contestant is assigned a door, then offered to switch.
Not without sacrificing the whole point of the problem. Either you allow the switch before a reveal to one other door, in which the starting conditions haven’t changed, the contestant still and a 1/3 chance, or you allow the switch from the first door to both other doors, which is mathematically the same, but destroys the point. The whole point is to demonstrate that people are not perceiving the effect of the switch as equalling giving the contestant both other doors. People don’t catch that, and so their understanding of the probabilities is wrong.
Sometimes explanations are poor. Sometimes the person who has the right answer has the wrong reason because he doesn’t understand himself. And it’s tricky to put into words.
Hope you’re right! But I saw more recently that he said he spends most of his time either watching relatively meaningless college basketball games or trying to figure out the nature of reality. This blurb seems to fall under the aegis of “figuring out the nature of reality” (but doing a poor job of it).
This doesn’t appear to make any sense. You haven’t defined CHANCE and PROBABILITY in any meaningful way.
We’ve covered it pretty thoroughly in this (and other) threads. In the standard Monty Hall puzzle, it always makes sense to switch because you have a better chance of winning that way.
