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Conditional Probability and the Monty Hall Problem

Common Sense & Intuition v Intellectual Toil

By JR, June 15, 2022

The truth may be puzzling. It may take some work to grapple with. It may be counterintuitive. It may contradict deeply held prejudices. It may not be consonant with what we desperately want to be true. But our preferences do not determine what’s true – Carl Sagan

The whole problem with the world is that fools and fanatics are always so certain of themselves, but wiser people so full of doubts - Bertrand Russell from“The Triumph of Stupidity,” 1933 [1]

Many people believe the only thing one needs to detect deceit, lies, error and bullshit is common sense. Sadly, common sense is not very common, especially in today’s distracted atomized postmodern cultural wasteland of religious superstition and fundamentalism, self-help pabulum, mindless marketing and other rubbish on corporate controlled media. The political and socio-economic dysfunction and rampant corruption of parasitic keyboard/mouse capitalism and the nonsense disseminated in cyberspace, especially on social media platforms add to the widespread existential angst, pessimism, nihilism and cynicism.

Literacy rates are appalling and innumeracy is the norm, especially basic understanding of probability theory as lotteries and online gambling are increasingly becoming popular. These obsessive activities are colossal wastes of time and money that entice the masses with false hope. Moreover, common sense - however defined - is not only not very common, but not very useful in ascertaining the truth.

Common sense, generally perceived as empirically based sound judgement based on intuition and a pragmatic approach grounded in old adages such as “think before you act” is not so reliable. Good thinking, after all, is hard work and most people are lazy and look for the easiest cognitive path. If common sense is the only tool you use to defend against marketing claims, religious obscurantism and other pervasive bullshit, you will find common sense not very effective. What is needed is insatiable curiosity and lifelong learning, scepticism, logic, a scientific outlook and understanding of logical fallacies and other cognitive traps. For many people motivated reasoning, confirmation bias and innumeracy are just three of the most common among literally hundreds of intellectual failings. Human minds are capable of accepting views of truth, reality and explanatory systems of metaphysics or epistemology that are satisfactory and satisfying to one’s common sense and emotions even though they are completely mistaken.

If you find learning unappealing, are reluctant to critique authorities and experts, strongly dislike finding out that you are wrong sometimes, are indifferent to supporting your beliefs and behavior with fact and reason, dislike critically evaluating your own beliefs and actions, have no desire to intelligently explain to others your reasoning, or are uninterested in focusing on details, skepticism and bullshit detection are not for you. Bullshit detection requires critical thinking for which there are hundreds of excellent books such as the recent book by Irish physicist David Robert Grimes called Good Thinking: Why Flawed Logic puts us all at Risk and How Critical Thinking Can save the World. You will also find it very useful to have at least one good book on deductive and inductive logical fallacies such as Bo Bennett’s Logically Fallacious: The Ultimate Collection of Over 300 Logical Fallacies (Academic Edition)

Critical thinking is a learned process of logical inquiry, questioning, skepticism, scientifically based fact-finding and hypothesis testing to produce evidence that will hopefully end in an explanatory theory to facilitate true belief and real knowledge. To act without these conditions of truth and knowledge sets one up for failure and disappointment. For example, how many people understand the miniscule probabilities of success involved in lotteries and gambling, stupid activities that were declared illegal in a more civilized period in the recent past? There are several examples of fallacies in probability theory such as the gambler’s fallacy and the classical birthday problem that defy intuition and what we call common sense.

Consider what has become known as the Monty Hall Problem, based on the former popular television show. For those too young to recall, I’ll attempt to explain:

The game is somewhat similar to the infamous shell game. Imagine you are on a game show in which you are presented with three doors. Behind one of the doors is a new luxury car and behind the other two are goats. The game show host asks you to pick a door. The host knows where the car is. Suppose you choose Door 1. To make the game more interesting, the host will always open one of the two remaining doors to reveal a goat since he knows which door the car is behind. Suppose in this case the host reveals a goat behind Door 2. The car must now be behind either Door I or Door 3. The host then asks you to make a final decision: stick with your original Door 1 or switch to Door 3. Does it matter? Most people, even many university math majors with graduate degrees such as a masters or PhD, will claim that, as in a coin flip, it’s 50-50 or even odds that the car is behind either Door 1 or Door 3. But is it? The answer is NO!

If you are ever presented with a three door dilemma, please switch! No matter how many trials (3 doors or one million) you are presented with, you should always switch. After all, you have information you did not have before Monty opened Door 2 revealing a goat.


Of course, this advice holds only when it is safe to assume that show host Monty will always offer contestants the opportunity to switch, regardless of their initial choice in doors. If the game-show host was mischievous by nature, he might make the offer to stick or switch only if contestants initially picked the door with the grand prize behind it – which would be a rigged situation - like all gambling, whether lotteries, casinos or the stock market.

In the aforementioned scenario, switching would then always lead to a goat and the game-show host could keep the big prize for the next show.

The false claim that it doesn’t matter if you stick or switch is false – total bullshit, despite the fact that the vast majority of people, including college math majors, think it is even odds - 50-50, as in a coin flip. Switching in fact increases your probability of winning the car by quite a wide margin; in fact it doubles your chances. By laying out the sample space (possible outcomes) below you ought to be able to see why it matters whether you switch or stick with your original door choice.

At the beginning of the game, you had a 33% (one third- actually 33.3333... %) chance of picking the car and a 67% (two thirds – actually 66.6666… %) chance of picking a goat. After you’ve made your first choice and the host opens another door to reveal a goat, you now have additional information about the remaining door but no further information about your original choice. There is still a 33% (one-third) chance there is a car behind your original door, but by switching there is now a 67% (two-thirds) chance there’s a car behind the remaining closed door 3. While there’s a chance that you originally chose the correct door, you will double the probability of winning if you switch to the remaining door. In other words, the switch strategy is the winning strategy because it wins if you chose the wrong door on your first try which, whether you know it or not, will have occurred two-thirds of the time. If you are like most people you still don’t think the decision to stick or switch matters. However, it most certainly does matter. To emphasize this scenario more graphically, suppose there are 100 doors with 99 goats, and 1 car. Let’s say you select Door 32 and the host opens 98 doors (except Door 32 of course) leaving Door 78 unopened. He reveals goats behind each of the 98 other doors. Are you still going to stick with Door 32, assuming it is equally likely to contain the car as does Door 78? In this case, the chances Door 32 has the car is only 1% but the chance that Door 78 has the car is 99%.

Intuition sometimes works and in situations that demand quick decisions, we have little choice. Read the excellent book Thinking, Fast and Slow by Daniel Kahneman.

There are many articles online regarding the Monty Hall Problem - including the story involving Marilyn vos Savant, a woman claimed to have the world’s highest recorded IQ (228).

Marilyn vos Savant and Monty Hall | AstroWright (psu.edu)


Bayes Rule






For alternative and more detailed explanations of the Monty Hall Problem:

Monty Hall problem - Wikipedia

Solving the Monty Hall Problem with Bayes Theorem | by GreekDataGuy | Towards Data Science

🐐 Monty Hall Problem Simulator

The Simple Question that Stumped EVERYONE Except Marilyn vos Savant - YouTube


[1] From Mortals and Others: Bertrand Russell's American Essays, 1931-1935, v.2, p.28. Eminent mathematician, philosopher and public intellectual, Russell has been one of my primary intellectual and ethical influences ever since I discovered him as a high school student. Here is the essay from 1933 “The Triumph of Stupidity”:

What has been happening in Germany is a matter of the gravest portent for the whole civilized world. Throughout the last hundred and fifty years, individual Germans have done more to further civilization than the individuals of any other country; during the latter half of this period, Germans, collectively, have been equally effective in degrading civilization. At the present day the most distinguished names in the world of learning are still German; the most degraded and brutal government is also German. Of the individual Germans whose work has caused Germany to be respected, some are in exile, some in hiding, and some have disappeared, their fate unknown. Given a few years of Nazi rule, Germany will sink to the level of a horde of Goths.
What has happened? What has happened is quite simple. Those elements of the population which are both brutal and stupid (and these two qualities usually go together) have combined against the rest. By murder, by torture, by imprisonment, by the terrorism of armed forces, they have subjected the intelligent and humane parts of the nation and seized power with the view of furthering the glory of the Fatherland.
What has happened in Germany may well happen elsewhere. The British Fascists are not as yet a large party, but they are growing rapidly, and if at any future time there should be danger of a Labor Government that meant business, they would win the support of most of the governing classes. Meanwhile, the British government of India is a form of Fascism, all the worse for being alien. The British in India, like the Hitlerites in Germany, can only govern by putting the best people in prison.
Brute force plays a much larger part in the government of the world than it did before 1914, and what is especially alarming, force tends increasingly to fall into the hands of those who are enemies of civilization. The danger is profound and terrible; it cannot be waved aside with easy optimism.
The fundamental cause of the trouble is that in the modern world the stupid are cocksure while the intelligent are full of doubt. Even those of the intelligent who believe that they have a nostrum are too individualistic to combine with other intelligent men from whom they differ on minor points. This was not always the case. A hundred years ago the philosophical radicals formed a school of intelligent men who were just as sure of themselves as the Hitlerites are; the result was that they dominated politics and that the world advanced rapidly both in intelligence and in material well-being.
It is quite true that the intelligence of the philosophical radicals was very limited. It is, I think, undeniable that the best men of the present day have a wider and truer outlook, but the best men of that day had influence, while the best men of this are impotent spectators. Perhaps we shall have to realize that skepticism and intellectual individualism are luxuries which in our tragic age must be forgone, and if intelligence is to be effective, it will have to be combined with a moral fervor which it usually possessed in the past but now usually lacks.
In this gloomy state of affairs, the brightest spot is America. In America democracy still appears well established, and the men in power deal with what is amiss by constructive measures, not by pogroms and wholesale imprisonment. After the defeat of the French Revolution, democracy; discredited by the reign of terror, re-conquered the world from America. Perhaps America is destined once more to save Europe from the consequences of its excesses. (10 May 1933)




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