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Monty Hall, Co-Creator and Host of 'Let's Make a Deal,' Dies at 96
Monty Hall, the genial host and co-creator of "Let's Make a Deal," the game show on which contestants in outlandish costumes shriek and leap at the chance to see if they will win the big prize or the booby prize behind door No. 3, died at his home in Beverly Hills, Calif., on Saturday. He was 96.
[...] "Let's Make a Deal" became such a pop-culture phenomenon that it gave birth to a well-known brain-twister in probability, called "the Monty Hall Problem." This thought experiment involves three doors, two goats and a coveted prize and leads to a counterintuitive solution.
[...] Mr. Hall had his proud moments as well. In 1973 he received a star on the Hollywood Walk of Fame. In 1988, Mr. Hall, who was born in Canada, was named to the Order of Canada by that country's government in recognition of the millions he had raised for a host of charities. In 2013 he was presented with a lifetime achievement award at the Daytime Emmys.
The Monty Hall problem:
Suppose you're on a game show, and you're given the choice of three doors: Behind one door is a car; behind the others, goats. You pick a door, say No. 1, and the host, who knows what's behind the doors, opens another door, say No. 3, which has a goat. He then says to you, "Do you want to pick door No. 2?" Is it to your advantage to switch your choice?
Vos Savant's response was that the contestant should switch to the other door. Under the standard assumptions, contestants who switch have a 2/3 chance of winning the car, while contestants who stick to their initial choice have only a 1/3 chance. [...] Many readers of vos Savant's column refused to believe switching is beneficial despite her explanation. After the problem appeared in Parade, approximately 10,000 readers, including nearly 1,000 with PhDs, wrote to the magazine, most of them claiming vos Savant was wrong. Even when given explanations, simulations, and formal mathematical proofs, many people still do not accept that switching is the best strategy (vos Savant 1991a). Paul Erdős, one of the most prolific mathematicians in history, remained unconvinced until he was shown a computer simulation demonstrating the predicted result.
Related: Get Those Brain Cells Working: The Monty Hall Problem
(Score: 4, Insightful) by theluggage on Monday October 02 2017, @02:30PM (6 children)
Nah. If Erdős didn't spot the "N doors as N->infinity" case then it ain't gonna convince the great unwashed. Even if there are N doors - I pick one, Monty reveals a goat, there are N-1 doors left, so the probability of the car being behind any of the doors -including the one I picked - is 1/(N-1) - right? Actually, that is a true statement - its what you'd get if you picked again at random - its just not the correct answer to the question being asked.
The first tricky bit is realising that - unless you act on the new information Monty is giving you - the first door you choose will never have anything other than a 1/N chance of winning, however many goats Monty reveals. The second tricky bit is realising that - because the total probability must be 1 - the probability for every other door must go up every time Monty reveals a goat. The "paradox" is, how can the universe distinguish between you forgetting what has gone before and picking again at random (prob of winning: 1/(N-1) ) and "switching" doors (prob of winning: ( 1 - 1/N)/(N-2) or whatever)
The whole thing is grist to the mill of whether probabilities have any meaning beyond predicting the outcome of a significant number of trials - here the "individual" case is so abstract and meaningless that it turns the problem into some sort of Schroedinger's Goat mindfuck. The thing that clarified it for me was not the result of doing a computer simulation, but simply trying to write one and being forced to think "what happens when X people do this?" which rapidly leads to "hey, this is a stupid question because the two cases are completely different". You have to assume that either everybody switches, or nobody switches, and in the last case you have a couple of lines of dead code.
Oh, PS, Obligatory XKCD [xkcd.com]
(Score: 0) by Anonymous Coward on Monday October 02 2017, @03:29PM (1 child)
Is it so complicated? When you first pick there is p = 1/3 chance it was correct (and hence q = 2/3 incorrect).
If p happened, there is 100% chance the host will offer you the wrong door to switch to. If q happened, there is 100% chance the host will offer you the correct door. So the probability of winning if you switch is just q*1 = 2/3.
(Score: 1, Interesting) by Anonymous Coward on Monday October 02 2017, @05:59PM
But remember, there are four cases for the rules:
1) The host only shows you a goat when you have picked the car.
2) The host only shows you a goat when you haven't picked the car.
3) The host always shows you a goat.
4) The host is capricious about showing you a goat (the real Monty Hall).
What trips people up is that they don't stick to a single case while they do their analysis. The problem statement makes it easy to fall into that trap.
(Score: 2) by fritsd on Monday October 02 2017, @05:46PM
Nicely explained!
And that, dear people, is why it's so bloody difficult to get your head around Bayesian statistics :-)
It just feels un-natural.
(I think)
But as Monty adds information, the prior changes. The probability density function changes from a flat uniform 1/N line to a series of taller block wave functions with small goats inbetween.
(Score: 2) by VLM on Tuesday October 03 2017, @04:14PM (2 children)
The car can't move on its own, and that darn Monty keeps shrinking the territory the car could be hiding in. Its not the universe messing with the odds, its Monty. Monty is a cheater and he knows where the car is and he's telling you where it isn't, which is almost the same as telling you where it is, once he's leaked enough info.
Maybe a hunting / fishing analogy. Ya got 100 ice fishing ponds and exactly one fish. The odds of you ice fishing on your first guess and catching the fish are really low. That darn Monty tosses dynamite into 98 of what he knows are empty ponds and nothing floats... "obviously" the fish isn't in your first guess pond, and it must be in the one remaining pond Monty didn't dynamite, right? The numbers scale all the way down to precisely three ponds. Or some similar weird analogy with deer hunting.
(Score: 2) by theluggage on Tuesday October 03 2017, @09:55PM (1 child)
Sure, but the slippery bit (that has fooled the great and the good) is getting why that hasn't changed the odds of the car being behind your door, but has changed the odds for the remaining door. The "total probability must be 1" thing proves it, but the trouble is that proving something like that is not the same as explaining something. What explained it for me was the experience of writing code to simulate it - which forces you to "depersonalise" the whole thing and think in terms of how repeated trials might work.
(Score: 2) by VLM on Wednesday October 04 2017, @07:43PM
Hmm well flip positive and negative thinking like I alluded to in another post.
I'm not saying I think there's a 1/3 chance the car is behind my door, I'm really saying there's a 2/3 chance its behind the other doors. And that pesky Monty leaked information such that "the other doors" has collapsed down from a buncha doors, down to precisely one remaining unopened unselected door. All the 2/3 is sitting on that one remaining door.
The 2/3 chance of it being behind other doors has remained constant from the first step. Its just the set of "other doors" has collapsed thanks to Monty from a buncha doors to precisely one door. That one remaining unopened unselected door owns the whole 2/3 chance subspace now, it doesn't share with multiple doors.
Its an English Language puzzle. Instead of all this "where is the car" odds, it seems clearer if we talk about "where isn't the car" odds. I've seen attempts at artificial languages from esperanto to lojban and it would be interesting to see mathematicians write a general purpose language. Well, interesting might not be the correct word. But it would be something, thats for sure.