{"id":696,"date":"2015-11-11T15:09:31","date_gmt":"2015-11-11T23:09:31","guid":{"rendered":"https:\/\/schoenblog.com\/?p=696"},"modified":"2015-11-11T15:09:31","modified_gmt":"2015-11-11T23:09:31","slug":"the-problem-with-the-monty-hall-problem","status":"publish","type":"post","link":"https:\/\/schoenblog.com\/?p=696","title":{"rendered":"The Problem with the Monty Hall Problem"},"content":{"rendered":"
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I was so taken with Leonard Mlodinow<\/a>‘s new book\u00a0The Upright Thinkers<\/a><\/em>, that I went on Amazon and ordered some of his previous books, including\u00a0The Drunkard’s Walk: How Randomness Rules Our Lives<\/a>.<\/em>\u00a0In an early chapter discussing the mathematical development of probability, Mlodinow discusses the Monty Hall problem<\/a>, made famous in 1990 by Marilyn vos Savant<\/a> in\u00a0Parade<\/a><\/em> magazine. \u00a0In\u00a0her\u00a0Ask Marilyn<\/a><\/em> column,\u00a0reproduced (with slight modifications) by Mlodinow, Savant was asked:<\/p>\n

Suppose the contestants on a game show are given the choice of three doors: Behind one door is a car; behind the others, goats. After a contestant picks a door, the host, who knows what’s behind all the doors, opens one of the unchosen doors, which reveals a goat. He then says to the contestant, “Do you want to switch to the other unopened door?” Is it to the contestant’s advantage to make the switch?<\/p><\/blockquote>\n

Marilyn answered “yes,” but thousands of her readers, including some with advanced degrees, wrote her to tell her she made a mistake<\/a>. \u00a0Marilyn solved the problem using elementary probability. \u00a0If you are given a choice of three doors, you will pick the correct one just 1\/3 of the time. \u00a0For the 2\/3 of the time that you initially chose incorrectly, you could win by switching your choice to the door that remains after the host reveals the first goat. \u00a0This seems a bit counterintuitive, which is why so many of her readers were upset, because you are still left choosing between two doors, not knowing what is behind either one. \u00a0It seems like a 50-50 choice.<\/p>\n

Many people, including Mlodinow, use this example to show how our intuition can be faulty and not correct according to the laws of probability. \u00a0Daniel Kahneman<\/a> won a Nobel Prize in Economics for this type of work. \u00a0But the problem with this example (and even\u00a0some\u00a0of Kahneman’s<\/a> that I have read) is that they\u00a0don’t really prove that our intuition is wrong.<\/p>\n

Take the Monty Hall problem. \u00a0Look at it more closely and consider this: how does the contestant know what game he is playing? \u00a0All he knows is that he was asked to choose one of three doors, then one of the unchosen doors was exposed and he was asked whether he wanted to switch. \u00a0That’s it. \u00a0The contestant has no way of knowing what game the host is actually playing. \u00a0And that makes all the difference for what our intuition tells us to do.<\/p>\n

It isn’t hard to imagine that this is not the game that Marilyn thought it was. \u00a0What if the game was this: if you choose incorrectly, the host opens the door to reveal your goat; but if you choose correctly, the host tries to induce you to change until you pick a door that has a goat or accept a smaller gift. \u00a0Sometimes he opens one of the unopened doors to reveal a goat and asks if you want to switch to the other unopened door (as in Marilyn’s example). Sometimes he asks you if you want cash instead of what is behind the door you chose. \u00a0Etc. Etc. \u00a0You never know if or how the game will end. \u00a0Now this game looks to you the contestant exactly like the game that Marilyn described. \u00a0If the host induces you to switch your choice, it means you’ve lost. \u00a0Do you want to follow Marilyn’s advice? \u00a0You’d lose every single time.<\/p>\n

So, back to our supposedly faulty intuition. \u00a0Is it really so bad? \u00a0What makes more sense: that the host would give us an easy chance to double our odds of winning, or that he would try to trick us into giving up our correct choice? \u00a0Our intuition tells us to be skeptical. \u00a0It tells us that we don’t have enough information to know what game we are playing, and so the choice is really 50-50. \u00a0We just don’t know if the information we received when the host revealed the goat really means anything. \u00a0So we are left again with choosing randomly\u00a0between two doors.<\/p>\n

While it is certainly true that our intuition often is at odds with the laws of probability, (and indeed that is why many people are initially confused by this problem), it turns out that the confusion is sometimes more warranted than strict probabilists are willing to admit. \u00a0There are plenty of real-world examples that show how bad people are at probability. \u00a0Take, for example, the fact that so many people buy lottery tickets, which are a terrible bet that no one in his right mind would take. \u00a0(In other words, the expected payoff is far less than the cost of a ticket.) \u00a0It makes you wonder though, why so many smart people like to use completely unreal problems, like the Trolley problem<\/a>, or flawed ones like the Monty Hall problem<\/a>, to make the same point. \u00a0You really cannot test the correctness of our real-world intuition with unreal make-believe problems where people can’t even know what game they are playing.<\/p>\n

[For a more thorough enumeration\u00a0of how\u00a0Marilyn might have been wrong when answering this problem, see Herb Weiner’s\u00a0Marilyn is tricked by a game show host<\/a>.<\/em>]<\/p>\n","protected":false},"excerpt":{"rendered":"

I was so taken with Leonard Mlodinow‘s new book\u00a0The Upright Thinkers, that I went on Amazon and ordered some of his previous books, including\u00a0The Drunkard’s Walk: How Randomness Rules Our Lives.\u00a0In an early chapter discussing the mathematical development of probability, … Continue reading →<\/span><\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"_monsterinsights_skip_tracking":false,"_monsterinsights_sitenote_active":false,"_monsterinsights_sitenote_note":"","_monsterinsights_sitenote_category":0,"footnotes":""},"categories":[1],"tags":[],"class_list":["post-696","post","type-post","status-publish","format-standard","hentry","category-uncategorized"],"_links":{"self":[{"href":"https:\/\/schoenblog.com\/index.php?rest_route=\/wp\/v2\/posts\/696","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/schoenblog.com\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/schoenblog.com\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/schoenblog.com\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/schoenblog.com\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=696"}],"version-history":[{"count":3,"href":"https:\/\/schoenblog.com\/index.php?rest_route=\/wp\/v2\/posts\/696\/revisions"}],"predecessor-version":[{"id":699,"href":"https:\/\/schoenblog.com\/index.php?rest_route=\/wp\/v2\/posts\/696\/revisions\/699"}],"wp:attachment":[{"href":"https:\/\/schoenblog.com\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=696"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/schoenblog.com\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=696"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/schoenblog.com\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=696"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}