# Let’s Make a Deal

This may be what Mueller is saying to Manafort, but I include those names just to try to trick google and get clicks.

I want to write about the very well known Monte Hall paradox. For the kids, there was this show “Let’s Make a Deal” featuring contestants and host Monte Hall who acted like a sleazy salesman trying to trick them. One constestant was the winner who got the final prize. They chose one of three doors. There as a big prize behind one of the doors. Monte would open another of the three doors showing a small prize, then he would ask the contenstant if he or she wanted to switch and take the prize behind the unchosen unopened door. I admit I watched the show. No one ever switched. Contenstants won the big prize about one third of the time (just as if they had guessed with no door opening or option to switch).

The funny thing is that if they had switched they would have won two thirds of the time. There were two doors left. The chance the big prize is behind the first door guessed by the contestant is 1/3 so the chance it is behind the unchosen unopened door is 2/3.

Bayes explains. Let’s say contestant chooses door 1 and Monte opens door 2. If the prize were behind door 3, he would open 2 with probability 1 (he always opened a door and the big prize was never behind it). so the probability that the prize was behind door 3 and Hall opens door 2 is (1/3)1 = =1/3 If it were behind door 1 he might open 1 or might open 3. *Assume* he chose at random in such cases. then the chance that the prize is behind door 1 and Hall opens door 2 is (1/3)/(1/2) = 1/6. So switching gives winning probability

(1/3)/((1/3)+(1/6))= 2/3.

This is extremely counterintuitive. Extremely. I remember advising my the contestants on my TV to not switch (I was very young). I recently learned Paul Erdos had to write a computer program to simulate the game to be convinced. That’s the Paul Erdos.

Why ?

1) loss aversion. People hate having something than losing it. Guessing right, then switching and losing is more painful than guessing wrong. I’m sure this is a factor, but it can’t beat two to one odds.

2) the cheater dectector (explanation due to John Tooby). During the whole show, if Monte Hall suggested doing something it was wise to say no. At the end, he operated according to simple rules. But it is natural to us to think “this man is trying to cheat me so I should say no). I think this might be it (being an actor Monte Hall managed to project more sleaze than Donald Trump himself). Very generally many paradoxes (or cases in which people don’t act as Bayes advised) make sense if one assumes that the experimenter might be lying. more on this below after the jump*

3) Our brains are hard wired to think about causation. The prize would be behind door 3 because 3 was chosen at random not because the contestent chose 1 and Hall opened 2. p is more likely if q is not the same as p is more likely because q and q caused p. This confuses us. We have trouble with correlation and causation. One fallacy is post hoc ergo propter hoc. Another is not propter hoc ergo no post hoc.

4. Did you see the stress on “assuming” ? Yes of course you did. The examples in which we should use Bayes formula and don’t always include strong assumptions about the data generating process (and Monte Hall US citizen, human being and free and equal agent was part of the data generating process). An older example is “I have 2 cards. one is black on both sides. one black on one side white on the other. I shuffle pick one and plop it down on the table. The upper side is black. The lower side is black with probability 2/3 not 1/2 as it is natural to guess. This works fine, because equal probabilities of one face up or the other make sense. Monte Hall is much more complicated. I can tell a story in which somone would have gained nothing by switching. Montey Hall opens the lowest number door which has not been chosen and doesn’t contain the large prize. If I choose 1 and he opens 2, then the probability that the big prize is behind door 3 is 1/2 (he opens 2 for sure if it is behind 1 or if it is behind 3). If he were to open door 3, then the probability that the prize is behind door 2 is 1. On average if one always swiches one wins 2/3 times but this is from 1/3 chance of winning for sure if one switches + 2/3 of winning half the time no matter what one does.

To calculate the probabilities, I need to know Monte Hall’s rule. I can tell (did tell) from watching let’s make a deal that he always opens a door and it never contains the big prize. But I don’t know he opened a door at random if the big priize is behind the first door the contestant chose. Now switching always gives at least as high probability of winning as staying, and the probabilities are only equal for a subset of measure zero of the manifold of may be Montes. But the sense that one needs to know something which one doesn’t know to do the calculations matters.

*more on cheater detectors

a) Allais paradox (cumulative prospect theory). People overweight low probabilities (Kahneman and Twersky) of extreme evends (Quiggin making sense of Kahneman and Twersky). 99% sure is,in our minds, much further from 100% sure than 89% sure is from 90% sure. Maurice Allais went to conferences on choice under uncertainty and managed to get people who had just given talks on expected utility maximization to make choices inconsistent with expected maximization of any function of winnings (choice which would be entirely rational if their sole aim in life were to amuse Allais). I have a rationalization. If someone claims something is 100% sure and it doesn’t happen, then there is proof beyond reasonable doubt that his claim was false. If he claims 99% there isn’t (notably lawyers, judges and jurors *really* can’t handle probability). This matters a lot in the real world where there is fraud and we think about whether it can be deterred.

b) Ellsberg paradox. there are 2 urns. One contains 50 purple balls and 50 orange balls. One contains 100 balls which are orange or purple. You can go to either urn and say “I win if I pull out a purple ball” or “I win if I pull out an orange ball” then pull out a ball and win if it is the color you said. The probability of winning if one choses the color at random is 50% for both urns. But we prefer the first urn. Why ? One explanation (Tooby type) is that the 50 and 50 statement is verifiable (by opening the urn). We prefer cases in which our counterparty says many things which can be proven to be lies. That makes trouble for crooks. The issue isn’t that we don’t know how many purple balls are in urn 2. The issues is that the experimenter knows and refused to tell us (ok refrained from telling us). In real life it is best to stear clear of such urns.

There are also fear of being lauged at (trying to pick a purple ball out of an urn with 100 orange balls is ridiculous). There is also the fundamental attribution errof of blaming people for bad luck. If I go to urn 2 and say purple and it has more orange balls, that is just bad luck. But I could have done better if I had guessed differently. That fact is painful.

I don’t think there is any door Mueller could offer than Manafort will take, and for a couple of reasons.

Health & Lifestyle

” He spent $6.4 million on real estate in New York and Virginia, another $7.3 million on home renovations and $820,000 on landscaping, prosecutors said in court documents. They said he paid $2 million for rugs in Virginia and antiques in New York. Nearly $1.4 million went to clothes at New York custom suitmaker Alan Couture and Beverly Hills designer House of Bijan, including a $21,000 Bijan “Royal Way” watch. (Bijan’s slogan: “The most expensive in the world.”) For three Range Rovers and a Mercedes-Benz, $300,000.

When Yanukovych fled Ukraine following a popular uprising in 2014, Manafort allegedly resorted to bank fraud to keep himself afloat, using his real-estate holdings to fraudulently secure $26 million in loans.

Manafort has pleaded not guilty to all charges.”

http://nymag.com/daily/intelligencer/2018/07/paul-manaforts-trial-everything-you-need-to-know.html

Anyone that spent like that is not going to give up any chance to reach that kind of income again.

In terms of health, people that piss off putin die on a regular basis and sometimes in irregular ways. Poison. Gun shots. Strangling. Defenestration.

He can’t take a deal.

He has to try and beat it; get sentenced; and hope for a pardon. Only chance he has.

The Monte Hall problem is interesting, but mathematically it only works if the presenter always shows you what’s behind one of the other doors. If he sometimes opens a door and sometimes doesn’t, he can manipulate the odds such that your odds of picking the right door are the same whether you switch or not (and he could actually make it lower odds of winning if you switch). Simple example: he only chooses to show you what’s behind one of the other doors if you’ve chosen the right door. Then you’re worse off to switch.

I believe Monte Hall didn’t always open a door, so you may have been right to hope the contestant didn’t switch.

Emike,

2014 “popular” uprising? You should define popular and what Vickie Nuland was in Kiev town supporting.

What is going on in Donbass/Donetz is a popular uprising, without US insigation!

“people that piss off a Clinton putin die on a regular basis” with as much evidence as the Putin meme.

Be thankful there isn’t a quantum mechanical Monte Hall problem. Electrons can have positive or negative spin, but if you have two electrons, the odds of both having positive spin are 1/3. The odds of both having negative spin are 1/3, and the odds of them having different spins is 1/3. Why? All electrons look alike. Even quantum mechanics can’t tell one electron from another, so there are only three possible states ++ — and +-.

One simply needs to imagine, say 100 doors, and Monty opens 98 of them and there is no prize in any of the 98. The probability that the contestant has picked the door with the prize is 1/100.

Welcome to Angry Bear. First time comments always go to moderation to weed out spammers and advertising.

Manafort situation now is difficult. But the crimes he is accused of were committed outside the election campaign period. He has some chances to fight them with a good lawyer team claiming the Mueller exceeded his mandate and engaged in the witch hunt against Trump.

If we assume the Mueller is a hired gun of Clinton wing of Democratic Party, and his appointment was a gambit to impeach Trump, then he is also in a difficult position.

1. Now a lot of people started raising unpleasant questions about his role in 911 cover-up. So he is investigated too.

2. After spending taxpayers money for more than a year, the results were questionable. He suffered greatly from Strzokgate and Steele dossier saga,

3. As Hillary aptly said” If that bastard wins, we all hang from nooses!” so I would assume that Trump digs out some skeletons too.

4. If Rosenstein falls, Mueller is cooked. There are some people who would like to take revenge, and without “Lord-protector” in the Justice Department, he is very vulnerable.

5. The direct interference of the intelligence agencies in the election and derailing Sanders now make all Russiagate saga a double-edged sword. There is also “the Sword of Damocles” over Dems due to Avan brothers scandal. Those can be played strategically.

So this catfight between two factions of the US neoliberal elite might be very interesting to watch.

In any case, Russiagate is just a smoke screen to cover the huge crack in the neoliberal state façade.

@Likbez,

You lost me at “If we assume the Mueller is a hired gun of Clinton wing of Democratic Party . . . ” You obviously know nothing about the Mueller investigation. One might just as well “assume” that Mueller is a sasquatch out of costume.

Smarter trolls, please.

Of course like many “gotcha” puzzles, the solution to the Monte Hall problem assumes facts that are NOT in the original question. This ASSUMES that Monte Hall knows which door the prize is behind., That he always CHOOSES the door with no prize behind it. It is just as possible that his choice of another door was random and that in this instance, he HAPPENED to open the door with no prize.

PJR Monte Hall always opened a door. Always. Every day for years. Contestants knew this.

JIm A. Hall knew where the big prize was. That’s how he could always open a door and it was never the door with the big prize behind it.

Monte Hall was (and I hope still is) a living human being who did things on television. People my age remember the show. We don’t include “always opened a door which never had the big prize behind it” any more than we mention that, on day, the sun rose in the East. It’s just something everyone knew.

Importantly the contestants knew this and they never switched doors (at least I never saw anyone switch). It’s all on video. We don’t need Bayes or Fisher — we can calculate actual sample frequencies. How many people won the big prize? I am 99% sure that it is less than 35% — Vishnu is subtle but He is not deceitful. How many would have won if they had switched doors. I am

~~100%~~99.999% sure* that it is 100% minus the fraction that actually won — Monte always opened a door and that never revealed the big prize) so I am 99% sure that the number is greater than 65%.* I typed 100% but, then again, I may be a butterfly dreaming that I am an angrybear. It’s possible that Monte messed up once. It’s possible that the Earth is flat.

I never watched the show but if, as you say, Monte Hall always opened a door with a minor prize, then he rendered that door irrelevant and the odds for the contestant become 50:50 whether he switches or not. There are two doors, the big prize is behind one of them and all of the posturing with respect to Hall’s opened door is meaningless as to affecting the chances.

A sighhh as much Mueller and Manafort as Monte.

ilsm a huge crowd in a public square is a popular uprising. Popular uprisers are not necessarily good guys. It is a statement about the number (large) and the approach (including the use of force including a trebuchet).

Indeed the troubles in Donbass & Donnetz began as protests involving the threat of force (a popular uprising). However, they later involved the Russian army invading another country (with uniforms but without insignia).

Vickie Nuland might not be popular (don’t know her) but her curiousity doesn’t change what Ukrainians were doing.

The statement “people that piss off a Clinton putin die on a regular basis” is ungrammatical (I think you meant to delete putin). It is also false, damaging, and written with reckless disregard for the truth.

You don’t have to worry. They won’t sue anyone ever. Also the Angrybearblog has no liability — we aren’t the author, editor or publisher of your libel.

But your comment is a tort.

@Likbez, what Joel said (with compliments for the topical reference to Virginian congressional campaigns). Mueller is a lifetime Republican appointed bt lifetime Republican Rod Rozenstrein who was appointed by sometimes Democrat Donald Trump.

The probability that “is a hired gun of Clinton wing of Democratic Party” is, like the probability that you are a butterfly, one of those cases which help us decide if we can believe that a probability can really be exactly exactly zero.

For that reason only, your comment is not off topic.

Robert,

We are too old. If you pose the problem now you have to state the assumptions because too many people do not know any more.

Bill H. One of the assumptions is that the contestant has already chosen a door before Monte opens one. If the contestant chase the big prize, it does not matter which Monte chooses, it will be a lesser prize. If the contestant chose a lesser prize (which happens 33% of the time), Monte has only one chose. The problem is one of conditional probability. Monte knows where the big prize is and his actions depend on the contestant’s actions. Even knowing all the assumptions, you want to believe that it is a simple coin flip, but not so.

Jim A

It was not always a gotcha question. It was a game show with real prizes. The same thing happened at the end of every show (I forget if this means 7 days a week or fewer). There were always 3 doors. The prizes were described. One was much larger than the other two. The contestant always chose one door. Monte Hall always opened another door showing one of the two small prizes. Monte Hall asked the contestant if he or she wished to switch and choose the door which had not yet been chosen or opened. The contestant said no. This happened day after day for years.

Almost all of the contestants knew all of this — they had watched the show many times before attending it as part of the studio audience and being told to “come on down” by Monte Hall.

The fraction of times the big prize was won is a matter of historical record. The fraction of times the big prize would have been won if the contestant switched doors is also a matter of historical record. I don’t know the numbers but I am willing to offer million to one odds that the contestants won the big prize less than 40% of the time and would have won it over 60% of the time if they had switched doors.

Anyone willing to put up 1 cent against $10,000 ? You will have to do the calculations (I don’t work for one penny) and the winner must be judged here by an angry bear.

It’s all on video.

Also the gotcha questions often getya. They trick us even if all the necessary assumptions are clearly stated. When young, I was sure it would be a mistake to switch. When I was 29 and had a PhD in Economics from Harvard U, I was gotme’d by my PhD supervisor who pulled Monte Hall on me.

But fuck me (not literally) Paul Erdos fell for it. The one and only Paul Erdos.

I would explain why the probabilities are 1/3 and 2/3 not 1/2 and 1/2 but I can’t do it as well as Arne.

@Robert Waldmann August 1, 2018 10:13 am

This is just a deflection. Nobody can deny that we observe a fight between two factions of the US elite. Which is about the direction of the country. Russiagate is just a smoke screen.

And Mueller actions talk louder than words, or this superficial detail of his resume (Democratic Party after Bill Clinton can well be renamed into Moderate Republican Party).

Look at the composition of Mueller team and try to find people who might be sympathetic to Trump platform (not that he lasted long; he betrayed it in three month in office). All the team consists exclusively of rabid Clinton supporters. Who knows what is their main task without the necessity of Mueller telling them anything. And as we all know “Personnel is policy.”

Now tell me again the he is a lifelong Republican 😉

Also being a Republican (and moreover, being the head of FBI after 911, and one of the architects of transition of the USA into national security state) does not exclude actions against detractors from neoliberal globalization and neoliberalism even if they are fellow Republicans.

His loyalty is not to the Republican party, but to neoliberalism and neoconservatism including neoliberal globalization, which is assaulted by Trump. Looks how smoothly neocons aligned with the Democratic Party during and after the elections.

Lickbenz I replied to your comment “If we assume the Mueller is a hired gun of Clinton wing of Democratic Party”. Now you assume that any connection with the Democratic Party is irrelevant. I assume that you conclude, as I do, that someone who introduced that organization in the discussion and then said it was irrelevant has nothing useful to say.

You brought up the “Democratic Party” not I. You now say parties are irrelevant distractions.

I will not reply to any reply to this reply to your reply to my reply to your comment.

@Robert

Still overthinking the problem. One door is open, two are closed. Contestant has chosen one of the two closed doors. Whether he chose it before Monte Hall opened the door or not is irrelevant. He has chosen one of two closed doors. His chances are precisely the same for either door, staying with his choice or switching.

His chances before Monte Hall opened the door would be different, but he is presented with that choice after the door is opened, at which time there are two closed doors and his chances are 50:50.

Bill H

The odds of winning if one chooses a door and doesn’t switch are one in three. They don’t change because Monte Hall opened another door and invited the contestant to switch.

The fraction of contestants who won the big prize was significantly less than 50%. Your claim can be tested with available data and is false. This is a matter of historical fact. It is all on video. Claiming that the probability is 50% is like claiming that the world is flat. There are relevant data which decide the question.

I haven’t checked them, but I have offerent 1,000,000 to 1 odds to anyone who does. Recall I bet $10,000 vs 1 cent that the fraction of winners of the big prize (on video) was less than 40%.

It is easy to simulate the game. Here is a gauss program which shows the probability of winning without swtiching is 1/3 and the probability of winning after switching is 2/3

num = 100000;

monte = rndu(num,1);

monte = 1+floor(3*monte);

guess = rndu(num,1);

guess = 1+floor(guess*3);

same = guess.==monte;

mch = rndu(num,1);

opn = (1-same).*((monte.>1.5).*(guess.>1.5)+3*(monte.<2.5).*(guess<2.5) + 2*(((monte-2).*(guess-2)).<0))

+ same.*((monte.==1).*(2+floor(2*mch) )+ (monte.==3).*(1+floor(2*mch))+ (monte.==2).*(1+2*floor(2*mch)));

switch = 1-same - (monte.==opn);

meanc(same);

meanc(switch);

“All the team consists exclusively of rabid Clinton supporters.”

More troll rubbish. Can we please have smarter trolls?

“I will not reply to any reply to this reply to your reply to my reply to your comment.”

Smartest thing you have ever said. I hope all of us treat this person in this manner.

Kind of lyrical? Anyhoo, he has made more intelligent statements.

“Contestant has chosen one of the two closed doors.”

No. Contestant chose one of three closed doors.

What happened after that depends on what he chose. If he chose a goat (the stated prize in the form I heard the puzzle), Monte was forced to leave the car behind the door to which the contestant can switch. If he chose the car, Monte can act randomly. The decision tree does not have equal weights, so the probability of the outcomes are not equal.

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