# Monty Hall problem

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The Mathematics of Banking and Finance by Dennis W. Cox, Michael A. A. Cox

However, if B or C is chosen, then Monty, respectively, must reveal C or B. These options are set out in Table 22.8. Simulation: Monte Carlo Methods 213 Table 22.7 Cumulative initial probabilities for the Monty Hall problem Choice Cumulative probability Probability A B C 0.3333 0.3333 0.3333 0 0.3333 0.6666 1 Table 22.8 Second event probabilities for the Monty Hall problem Initial choice Shown Probability Stick Switch B C C B 0.5 0.5 1 ! A A A A C B B C A A B C Table 22.9 Simulation of the Monty Hall problem Random variable 0.8444 0.1011 0.5444 0.9736 0.4440 0.2745 0.9083 0.6326 0.5568 0.6940 Initial choice Random variable Shown C A B C B A C B B C 0.8691 0.0009 0.4648 0.2184 0.7167 0.8878 0.4055 0.0617 0.9972 0.9833 B B C B C C B C C B Switch A C A A A B A A A A win lose win win win lose win win win win Stick C A B C B A C B B C lose win lose lose lose win lose lose lose lose Table 22.10 Summary of a Monte Carlo simulation of the Monty Hall problem Switch Stick Win Lose 1,320 680 680 1,320 Ratio 0.66 0.34 The player may either stick with the original choice or switch.

Initial choice Shown New choice 0.5 A 0.5 B 0.0833 No change & Win 0.0833 Change & Lose 0.0833 No change & Win 0.0833 Change & Lose 0.1667 Change & Win 0.1667 No change & Lose 0.1667 Change & Win 0.1667 No change & Lose 0.5 C 0.3333 A 0.5 A 0.5 C 0.5 B 0.5 A 0.3333 B 1.0 C 0.5 B 0.5 A 0.3333 C 1.0 C 0.5 C Figure 4.9 Tree diagram for the Monty Hall problem. Probability Theory 35 Table 4.2 Summary probabilities for Monty Hall problem Change No change Win Lose 0.3333 0.1667 0.1667 0.3333 Therefore in order to maximise the chance of winning, the optimal strategy is for the contestant to change his choice, since he is twice as likely to win than if he fails to switch. Monty has shown the contestant additional information by opening one of the doors, which has provided the contestant with a better chance of winning.

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Radical Uncertainty: Decision-Making for an Unknowable Future by Mervyn King, John Kay

The dial’s first move records the probability that A will win the match given that he has won the first game , and then subsequently adjusts to the probability that he will win overall conditional on A having won the first game, but B having won the second , and so on as the evening progresses. Hall The Monty Hall problem 15 is a famous illustration of the power of Bayes’ theorem, loosely based on the 1960s American quiz show Let’s Make a Deal , in which contestants would bid for prizes hidden behind curtains, and named after its host. The puzzle was originally posed by the American statistician Steven Selvin – and has been subsequently the subject of extensive correspondence and literature.

The Indifference Principle The solutions to the problem of points and the Monty Hall game rely on what has become known as the Indifference Principle – that if we have no reason to think one thing more likely than another, we can attach equal probabilities to each. We assumed that the Duke and Marquis were equally likely to win each of the remaining games, and perhaps there was a frequency distribution of past results of similar games to guide our conjecture. In the Monty Hall problem, we judged that if there were three identical boxes the probability that the keys were in any one of them was one third. 17 John Maynard Keynes is known to everyone for his many contributions to public policy in Britain and internationally during the inter-war period and the Second World War.

(In Let’s Make a Deal , raucous audience participation was an essential part of the show’s now difficult to understand appeal to its viewers.) But once everyone understands the problem, then how can the show sustain its interest? Can the viewers be confident that the original rules are still being applied? Real worlds are always complex. Many commentators and teachers use the Monty Hall problem to emphasise that a puzzle, or model, can only be ‘solved’ if the assumptions made are completely specified. And this observation is correct. But in a world of radical uncertainty, problems are rarely completely specified. The mathematics of probability requires that the sum of the probabilities of all possible events adds up to 1.

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The Drunkard's Walk: How Randomness Rules Our Lives by Leonard Mlodinow

14 No, it is not: If the students were lying, the correct probability of their choosing the same answer is 1 in 4 (if you need help to see why, you can look at the notes at the back of this book).15 And now that we’re accustomed to decomposing a problem into lists of possibilities, we are ready to employ the law of the sample space to tackle the Monty Hall problem. AS I SAID EARLIER, understanding the Monty Hall problem requires no mathematical training. But it does require some careful logical thought, so if you are reading this while watching Simpsons reruns, you might want to postpone one activity or the other. The good news is it goes on for only a few pages. In the Monty Hall problem you are facing three doors: behind one door is something valuable, say a shiny red Maserati; behind the other two, an item of far less interest, say the complete works of Shakespeare in Serbian.

In the words of a Harvard professor who specializes in probability and statistics, “Our brains are just not wired to do probability problems very well.”7 The great American physicist Richard Feynman once told me never to think I understood a work in physics if all I had done was read someone else’s derivation. The only way to really understand a theory, he said, is to derive it yourself (or perhaps end up disproving it!). For those of us who aren’t Feynman, re-proving other people’s work is a good way to end up untenured and plying our math skills as a checker at Home Depot. But the Monty Hall problem is one of those that can be solved without any specialized mathematical knowledge. You don’t need calculus, geometry, algebra, or even amphetamines, which Erdös was reportedly fond of taking.8 (As legend has it, once after quitting for a month, he remarked, “Before, when I looked at a piece of blank paper my mind was filled with ideas.

But unless you can bend silver spoons into pretzels with your brain waves, the odds are 2 to 1 that you are in the Wrong Guess scenario, and so it is better to switch. Statistics from the television program bear this out: those who found themselves in the situation described in the problem and switched their choice won about twice as often as those who did not. The Monty Hall problem is hard to grasp because unless you think about it carefully, the role of the host, like that of your mother, goes unappreciated. But the host is fixing the game. The host’s role can be made obvious if we suppose that instead of 3 doors, there were 100. You still choose door 1, but now you have a probability of 1 in 100 of being right.

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The Wisdom of Finance: Discovering Humanity in the World of Risk and Return by Mihir Desai

New York: Basic Books, 2008; Kaplan, Michael, and Ellen Kaplan. Chances Are—: Adventures in Probability. New York: Viking, 2006; and Stigler, Stephen M. The Seven Pillars of Statistical Wisdom. Cambridge, MA: Harvard University Press, 2016. An excellent discussion of the Monty Hall problem can be found at https://www.khanacademy.org/math/precalculus/prob-comb/dependent-events-precalc/v/monty-hall-problem. On Buffett, see Frazzini, Andrea, David Kabiller, and Lasse H. Pedersen. Buffett’s Alpha. NBER Working Paper no. 19681, December 16, 2013. National Bureau of Economic Research. http://www.nber.org/papers/w19681.pdf; and Ng, Serena, and Erik Holm.

This question rapidly led to a discussion of the empirical regularity that in many countries there are slightly more boys born than girls, and this regularity is not fully attributable to the prevalence of abhorrent means of selecting boys over girls in some cultures. Twenty minutes into this discussion, we were talking about sex ratios, selective abortions, and infanticide—and no one had learned anything about probabilities. My second failed attempt was an effort to introduce the “Monty Hall” problem. In this problem, you are the contestant on the game show Let’s Make a Deal who gets to select one of three curtains. One of the curtains conceals a worthy prize while the other two curtains conceal booby prizes. After you select a curtain, the host of the show, Monty Hall, reveals that one of the remaining two curtains concealed a booby prize.

.), 122–23 mergers, 8 AOL–Time Warner merger mistakes, 109–12 asymmetric mergers (bolt-on acquisitions), 111 Ford Motor and Firestone Tire partnership, 117–18 General Motors and Fisher Body merger, 113–17 Hewlett Packard and Autonomy acquisition, 109 integration planning, 110 mergers of equals, 111 serial acquirers, 111 synergies, 109–10 Merton, Robert, 40 Miller, Alice, 95 Milton, John, xi, 7, 59, 68–69, 74 “When I Consider How My Light Is Spent,” 70–71 Miracle Worker, The (film, play), 96 Miranda, Lin-Manuel, 75 Molho, Anthony, 103 Monte Dei Paschi di Siena, 100 Monte delle doti, 101–4 “Monty Hall” problem, 16 moral hazard, 28–30 Morris, Robert, 142 career, 143–45 financier of the revolution, 143 relative to J. P. Morgan and John D. Rockefeller, 143 Mostel, Zero, 75 Moyers, Bill, 155, 158 N Newman, John Henry, 2 Newton, Isaac, 40 Nichols, Mike, 97 Nicomachean Ethics (Aristotle), 55 Nietzsche, Friedrich, 6 1984 (Orwell), 127 normal distribution, 19–20, 39 Nussbaum, Martha, 154–56, 158–60 O O Pioneers!

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Statistics hacks by Bruce Frey

Suffering from white-line fever? No, you've just applied the statistical solution to what is known as the Monty Hall problem and chosen the road among the three that has the greatest chance of being correct. Hard to believe? Read on, my friend, and prepare to win riches beyond your wildest dreams. The best strategy in this case is so counterintuitive and downright weird that the world's smartest people have disagreed aggressively about whether it even really is the best strategy. But believe meit is. The Monty Hall Problem and Game Show Strategy In our example with the three roads and the prospector, there is, in fact, a two-thirds (about 67 percent) chance that C is the correct road.

The 67 percent likelihood now transfers to curtain C. That's why you should always switch to the other curtain. If you were given the option of swapping your pick of one curtain for both the other two curtains, you'd switch in a second wouldn't you? That's essentially what is offered in the Monty Hall problem. Some figures might be necessary to persuade your inner skeptic. Look at Table 5-1, which shows the probability breakdown for the three options at the start of the game. You have a one-third chance of guessing the winning curtain and a two-thirds chance of picking a nonwinning curtain.

You might be wrong, of course, but you have a better shot of winning that car or whatever other prize you are playing for if you accept any offers to switch. This is always the best strategy, if a few criteria are met: The host knows what is behind each curtain. The host reveals one of the unchosen curtains and the prize is not behind it. Your original choice was random. The Controversy The Monty Hall problem and the general game show strategy that resulted was first introduced to the masses in 1991 by Marilyn Vos Savant, a columnist for Parade Magazine. Because she is known for being a "high IQ genius," Vos Savant answered questions from readers, sometimes of a brain teaser nature. Someone sent in the problem as I've described it, and she published the answer I have given here.

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When Einstein Walked With Gödel: Excursions to the Edge of Thought by Jim Holt

In a slightly ludic way, this essay bridges the fields of metaphysics, epistemology, and ethics, lending the volume a unity that I hope is not wholly specious. And lest I be accused of inconsistency, let me (overconfidently?) express the conviction that the “Copernican principle,” “Gödel’s incompleteness theorems,” “Heisenberg’s uncertainty principle,” “Newcomb’s problem,” and “the Monty Hall problem” are all exceptions to Stigler’s law of eponymy (vide p. 292). J.H. New York City, 2017 PART I The Moving Image of Eternity 1 When Einstein Walked with Gödel In 1933, with his great scientific discoveries behind him, Albert Einstein came to America. He spent the last twenty-two years of his life in Princeton, New Jersey, where he had been recruited as the star member of the Institute for Advanced Study.

It was winter and Gödel had an electric heater and had his legs wrapped in a blanket. I said, ‘Professor Gödel, what connection do you see between your incompleteness theorem and Heisenberg’s uncertainty principle?’ And Gödel got angry and threw me out of his office.” Overconfidence and the Monty Hall Problem Can you spot a liar? Most people think they are rather good at this, but they are mistaken. In study after study, subjects asked to distinguish between videotaped liars and truth tellers have performed miserably at the task, scoring little better than chance. That goes even for those who were especially sure of their expertise in catching out lies—police detectives, for instance.

He was also one of the world’s leading experts on probability theory; indeed, something he invented called the probabilistic method is often simply referred to as the Erdős method—thus making his name synonymous with probability. In 1991, Erdős found himself befuddled when the Parade magazine columnist Marilyn vos Savant published a probability puzzle called the Monty Hall problem, named after the original emcee of the TV game show Let’s Make a Deal. It goes like this. There are three doors onstage, labeled A, B, and C. Behind one of them is a sports car; behind the other two are goats. You get to choose one of the doors and keep whatever is behind it. Let’s suppose you choose door A.

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Evil by Design: Interaction Design to Lead Us Into Temptation by Chris Nodder

Journal of Personality and Social Psychology 99.5 (2010): 771–784. Higher social classes are more selfish: Jennifer E. Stellar, Vida M. Manzo, Michael W. Kraus, and Dacher Keltner. “Class and compassion: Socioeconomic factors predict responses to suffering.” Emotion 12.3 (2012): 449–459. Learning from casinos Monty Hall Problem: en.wikipedia.org/wiki/Monty_Hall_problem. Lottery sales and gambling income data: North American Association of State and Provincial Lotteries. Lottery Sales and Profits (naspl.org). 60 percent of adults report playing at least once per year: National Gambling Impact Study Commission staff-generated report on lotteries (1999). 72 percent of all gambling: The majority of gambling income comes from casinos (41 percent) and lotteries (31 percent).

It stays at 2/3 even after the host opens the right door. By switching, you keep the 2/3 probability with the added benefit of having one of the options eliminated. If that explanation made your brain hurt, welcome to the world that most people inhabit. The previous example is called the Monty Hall problem after the host who popularized it on the Let’s Make a Deal game show, and yes, they really did use goats. It is a great demonstration of the difference between mathematical probability and common sense. And yet strangely, despite the confusion that most people feel when dealing with questions of probability, they feel qualified to play the odds every time they enter a casino or buy a lottery ticket.

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The Moral Landscape: How Science Can Determine Human Values by Sam Harris

If you stick with your initial choice, however, your odds of winning are actually 1 in 3. If you switch, your odds increase to 2 in 3.54 It would be fair to say that the Monty Hall problem leaves many of its victims “logically dumbfounded.” Even when people understand conceptually why they should switch doors, they can’t shake their initial intuition that each door represents a 1/2 chance of success. This reliable failure of human reasoning is just that—a failure of reasoning. It does not suggest that there is no correct answer to the Monty Hall problem. And yet scientists like Joshua Greene and Jonathan Haidt seem to think that the very existence of moral controversy nullifies the possibility of moral truth.

He notes that when asked to justify their responses to specific moral (and pseudo-moral) dilemmas, people are often “morally dumbfounded.” His experimental subjects would “stutter, laugh, and express surprise at their inability to find supporting reasons, yet they would not change their initial judgments …” The same can be said, however, about our failures to reason effectively. Consider the Monty Hall Problem (based on the television game show Let’s Make a Deal). Imagine that you are a contestant on a game show and presented with three closed doors: behind one sits a new car; the other two conceal goats. Pick the correct door, and the car is yours. The game proceeds this way: Assume that you have chosen Door #1.

See also health memes, 20–21 memory, 116, 212n71, 234n54 Mill, John Stuart, 5, 199n10, 207n12 Miller, Geoffrey, 56 Miller, Kenneth, 173, 237n97 Miller, William Ian, 215n93 mind, 83–85, 110, 119, 158–59, 180. See also brain science; brain structures; theory of mind misogyny, 43, 196n9 Moll, Jorge, 91–92, 212n64, 213n78 Monty Hall Problem, 86, 211–12n54 Mooney, Chris, 174–76 Moore, G. E., 10, 12, 196n16 moral brain, 91–95 moral experts, 36, 198n6, 202n17 moral landscape: Bad Life and Good Life in, 15–21, 38–42 facts and values in, 10–14 flawed conceptions of morality and, 53 importance of belief and, 14 meaning of, 7–10 moral progress and, 177–79, 188, 191 problem of religion and, 2, 22–25 suffering and, 21–22 See also morality; values; and headings beginning with moral moral law, 33, 38, 161, 169–70 moral paradox, 67–77 moral persuasion, 49–50 moral philosophy, 81–82, 197–98n1, 199n10, 225n35.

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Are You Smart Enough to Work at Google?: Trick Questions, Zen-Like Riddles, Insanely Difficult Puzzles, and Other Devious Interviewing Techniques You ... Know to Get a Job Anywhere in the New Economy by William Poundstone

Poundstone, William. How Would You Move Mount Fuji? Microsoft’s Cult of the Puzzle: How the World’s Smartest Companies Select the Most Creative Thinkers. New York: Little, Brown, 2003. Selvin, Steve. “A Problem in Probability.” Letter to the editor. American Statistician 29 (1975): 67. ———. “On the Monty Hall Problem.” Letter to the editor. American Statistician 29 (1975): 134. Stone, Dianna L., and Gwen E. Jones. “Perceived Fairness of Biodata as a Function of the Purpose of the Request for Information and Gender of the Applicant.” Journal of Business and Psychology 11 (1997): 313–23. Thaler, Richard.

“have important applications in the cheese and sugarloaf industries”: Quoted in Gardner, The Scientific American Book of Mathematical Puzzles and Diversions, 34. Putzer and Lowen’s 1958 publication was a research memorandum issued by Convair Scientific Research Laboratory, San Diego. Selvin argued that you should switch boxes: Selvin, “A Problem in Probability.” had to defend it in a follow-up letter: Selvin, “On the Monty Hall Problem.” “has been debated in the halls of the Central Intelligence Agency”: Tierney, “Behind Monty Hall’s Doors.” only 12 percent of those questioned: Granberg and Brown, “The Monty Hall Dilemma,” 711. “Certainly Monty Hall knows”: Selvin, “A Problem in Probability.” “I wouldn’t want to pick the other door”: Granberg and Brown, “The Monty Hall Dilemma,” 718.

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The Man Who Solved the Market: How Jim Simons Launched the Quant Revolution by Gregory Zuckerman

The fund’s portfolio was market neutral, impervious to the overall stock market’s ups and downs. D. E. Shaw embraced a different hiring style than Renaissance. In addition to asking specific, technical questions about an applicant’s field of expertise, the firm challenged recruits with brainteasers, situational mathematical challenges, and probability puzzles, including the famed Monty Hall problem, a brain teaser based on the old television show Let’s Make a Deal. Employees, many of whom were fans of the British science-fiction television show Doctor Who, dressed informally, breaking Wall Street’s stiff mold. A 1996 cover story in Fortune magazine declared D. E. Shaw “the most intriguing and mysterious force on Wall Street . . . the ultimate quant shop, a nest of mathematicians, computer scientists, and other devotees of quantitative analysis.”

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Rationality: What It Is, Why It Seems Scarce, Why It Matters by Steven Pinker

But this is not how the problem works. Imagine that after you place your bet on #1, God announces, “It’s not going to be horse #3.” He could have warned against horse #2 but didn’t. Switching your bet doesn’t sound so crazy.42 In Let’s Make a Deal, Monty Hall is God. The godlike host reminds us how exotic the Monty Hall problem is. It requires an omniscient being who defies the usual goal of a conversation—to share what the hearer needs to know (in this case, which door hides the car)—and instead pursues the goal of enhancing suspense among third parties.43 And unlike the world, whose clues are indifferent to our sleuthing, Monty Almighty knows the truth and knows our choice and picks his revelation accordingly.

My Erdös number is 3, thanks to Michel, Shen, Aiden, Veres, Gray, The Google Books Team, Pickett, Hoiberg, Clancy, Norvig, Orwant, Pinker, Nowak, & Lieberman-Aiden 2011. The computer scientist Peter Norvig has coauthored a report with fellow computer scientist (and Erdös coauthor) Maria Klawe. 38. To be fair, normative analyses of the Monty Hall dilemma have inspired voluminous commentary and disagreement; see https://en.wikipedia.org/wiki/Monty_Hall_problem. 39. Try it: Math Warehouse, “Monty Hall Simulation Online,” https://www.mathwarehouse.com/monty-hall-simulation-online/. 40. Such as Late Night with David Letterman: https://www.youtube.com/watch?v=EsGc3jC9yas. 41. Vazsonyi 1999. 42. Suggested by Granberg & Brown 1995. 43.

In fact, before I knew about Arcsine Law, I made a mistake that I now see others making from time to time. This one is crazy complicated, so if you don’t understand the reasoning, just at least make sure you remember the conclusion. Arcsine Law is about as annoying and difficult to understand as the Monty Hall problem. But worth the effort to understand. First, let me reprint some faulty analysis I did in 2017. If you know why it’s wrong before you read the whole thing, you are either a genius or you already know about Arcsine Law. The following is an excerpt from a daily wrote toward the end of 2017. Have you ever noticed that the yearly highs and lows in G7 FX tend to happen more at the start and end of the year (and less in the middle)?

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Time Paradox by Philip G. Zimbardo, John Boyd

Tversky, “Variants of Uncertainty,” Cognition 11: 143–57 (1982). 19. D. Kahneman and A. Tversky, “The Psychology of Preferences,” Scientific American 246: 160–73 (1982). 20. Gilovich and Medvec, “The Experience of Regret.” 21. T. Gilovich, V. H. Medvec, and S. Chen, “Omission, Commission, and Dissonance Reduction: Overcoming Regret in the Monty Hall Problem,” Personality and Social Psychology Bulletin 21: 182–90 (1995). 22. H. B. Gerard and G. C. Mathewson, “The Effects of Severity of Initiation on Liking for a Group: A Replication,” Journal of Experimental Social Psychology 2: 278–87 (1966). 23. P. G. Zimbardo, “Control of Pain Motivation by Cognitive Dissonance,” Science 151: 217–19 (1966). 24.

The Book of Why: The New Science of Cause and Effect by Judea Pearl, Dana Mackenzie

Causal paradoxes shine a spotlight onto patterns of intuitive causal reasoning that clash with the logic of probability and statistics. To the extent that statisticians have struggled with them—and we’ll see that they whiffed rather badly—it’s a warning sign that something might be amiss with viewing the world without a causal lens. THE PERPLEXING MONTY HALL PROBLEM In the late 1980s, a writer named Marilyn vos Savant started a regular column in Parade magazine, a weekly supplement to the Sunday newspaper in many US cities. Her column, “Ask Marilyn,” continues to this day and features her answers to various puzzles, brainteasers, and scientific questions submitted by readers.

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Rationality: From AI to Zombies by Eliezer Yudkowsky

In this case, you know that all 7 other flips came up tails, and the posterior odds are 1:16 against the coin being H-biased. I could have decided in advance to say the result of the 4th, 6th, and 9th flips only if the probability of the coin being H-biased exceeds 98%. And so on. Or consider the Monty Hall problem: On a game show, you are given the choice of three doors leading to three rooms. You know that in one room is \$100,000, and the other two are empty. The host asks you to pick a door, and you pick door #1. Then the host opens door #2, revealing an empty room. Do you want to switch to door #3, or stick with door #1?

If the host always opens door #2 regardless of what is behind it, #1 and #3 both have 50% probabilities of containing the money. If the host only opens a door, at all, if you initially pick the door with the money, then you should definitely stick with #1. You shouldn’t just condition on #2 being empty, but this fact plus the fact of the host choosing to open door #2. Many people are confused by the standard Monty Hall problem because they update only on #2 being empty, in which case #1 and #3 have equal probabilities of containing the money. This is why Bayesians are commanded to condition on all of their knowledge, on pain of paradox. When someone says, “The 4th coinflip came up heads,” we are not conditioning on the 4th coinflip having come up heads—we are not taking the subset of all possible worlds where the 4th coinflip came up heads—rather we are conditioning on the subset of all possible worlds where a speaker following some particular algorithm said “The 4th coinflip came up heads.”