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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: 2) by maxwell demon on Tuesday October 03 2017, @07:01AM
That depends on how you generalize to more than three doors. You are probably thinking of the generalization "the host opens all remaining doors except for one" which is usually used for helping intuition.
But from the original problem description, the host opens another door, that is, one door, therefore the obvious generalization if not having a specific goal in mind is that also in the generalized version the host opens just one door. In that case, with more doors your chances go down with more doors. Indeed, the probability when changing then is (n-1)/(n(n-2)), which, while still better than 1/n, goes closer to it the more doors there are, so (as one would intuitively expect) the help you get by the host opening that door diminishes for large n. Note that no matter whether you switch or not, the probability of getting the car approaches zero.
Another possible generalization is that the host opens exactly half of the remaining doors (this of course only works if the total number of doors is odd). In that case, your chance when switching is 2/n, so you double your chances by switching, just as in the original problem, but your chances to get the car go to zero for large n, not to 1.
Note that all three generalizations reduce to the original Monty Hall problem for n=3.
The Tao of math: The numbers you can count are not the real numbers.