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Bayesian probability and the Monty Hall problem

On a rare slow Friday, I found myself in a musing of the Monty Hall problem.

You know the famous brainteaser, a probability puzzle in which you are a player in the game of guessing in order to win a prize. There are three doors closed, and behind each door is either a goat or a car, and whatever you have picked will be the prize for you. Of course, you want the car, not the goat. But there are two goats and just one car. So if you randomly pick, your winning chance is 1/3, a simple enough probability that almost everyone understands.

But here’s the trick. The host knows what is behind each door (this is important). He allows you to pick your choice first and tell him of your choice (this is also important). Then, instead of opening the door of your choice to end the game (which would be incredibly boring), he then picks one of the other two doors you didn’t choose and shows that there is a goat behind that door. The host then asks you, “Do you wish to stick with your choice or to change it?”

Let me just jump to the answer instead of working you the reader too much: You absolutely want to switch to the other door because, at that moment, if you don’t switch, your winning chance is 1/3, but if you do, your winning chance is 2/3.

This has been discussed to death in numerous writings, but I will provide my simple explanation later, which is not the main point I wish to make here (as there’s also another high-level hint that can be learned from Monty Hall problem, see below). But I’m quite sure my explanation makes more sense than any other ones you’ve ever seen before.

On a related note, if you are interested in Blockchain and Bitcoin, I hope you may read the article on applying Bayes’ theorem to the question of the identity of Satoshi, the inventor of Bitcoin. See Mathematical proof that Craig S. Wright is Satoshi Nakamoto.

If, after reading both the present article and the article linked above, you don’t see any connection between determining the identity of Satoshi and the Monty Hall problem, you’re probably missing the point I’m making in this article.


It is noteworthy that when the Monty Hall problem was published, an overwhelming majority of people got it wrong. Most people believe it does not matter whether you change your choice or not; the probability remains the same because after the host has picked one of the two goats, the remaining two doors (including the one the player has initially picked) have one goat and one car, so the probability is half-and-half, and there is no point to switch.

Most people probably follow mere intuition (which is, unfortunately, wrong; see below). Still, many people who are trained academically not only get it wrong, but they would get upset with the right answer because they absolutely believe that the answer is wrong based on their knowledge of abstract mathematical probability, which cannot possibly be wrong. It’s an outrage to challenge it using a ‘trick’. If you read the comments made by thousands of PhDs at the time, you will have a feeling of what I’m talking about here.

But mistakenly considering each point of time as an independent coordinate in space-time is precisely why people get it wrong, and also why educated people are more likely to get it wrong because they are trained to analyze things more deeply and more exactly, but more likely in that wrong way.

The world is connected and keeps updating with more and more information; some are signals, but more are just noise. The ability to think independently is necessary to eliminate noise, but ultimately, the ability to estimate each situation in a connected way is also necessary to reach the right conclusion because otherwise, you misread the signals by giving them incorrect relevance.

In other words, maintain the independence of your mind from other people’s opinions and influences, but always remember the dependence of every situation on other events in this sequential and consequential world. This is not dialectic but an essential definition of a valid human observer of the universe, a unique part of the entire creation. If one loses the independence of his mind yet fails to realize the consequential nature of the world, or vice versa, his thought is more likely to be noise than signal to others.

A main revelation of my musings today is that Monty Hall problem wasn’t merely a clever game but a test of people and society’s ability to evaluate the available evidence in order to determine the probability of a truth (or a lie).

The key is that we naturally think about probability as if it were a simple outcome of evidence determined in one collective estimate. In other words, when it comes to probability, we tend to think that we live in a world in which events are independent of each other, and thus allow our brain to estimate the probability by taking a single snapshot of the whole universe.

We think linearly. Our brain muscles are not accustomed to thinking non-linearly. One evidence for this is the fact that most people don’t intuitively understand the effect of compounding effect in investment. Most can understand the concept itself once it is explained, but our brain still doesn’t naturally follow the force of compounding. The compounding effect is only one of the simplest sequential and consequential relations in this world.

Bayes’ Theorem

Put in another way. We do not know how to properly estimate the probability according to Bayes’ Theorem, which is the law of probabilities and tells us how to evaluate a probability (which is an expression of the degree of belief or confidence in a truth) in a world of relatedness and continuity, by continuously updating the posterior probability.

More specifically, Bayes’ Theorem teaches us how to rationally update a degree of belief (expressed in a probability) by taking into account all available related evidence, not all at once as a single snapshot, but in a consecutive manner with respect to the dependencies on space and time.

Our universe is not a mere structure. It is a story.

Once we start to understand the relatedness and continuity of the world in which we live, our rational conclusion can be very different from our gut feeling. Gut feeling is important, especially when there is no direct evidence to our eyes, but following the gut feeling is also dangerous when you are buried with too much information (all purporting to be evidence) and your gut is inundated, dulled, and eventually misled.

Be rational mentally about material matters and rational spiritually with spiritual matters. Rationality is an essential element of honesty and integrity.

For an example of applying Bayes’ theorem in real life, see Mathematical proof that Craig S. Wright is Satoshi Nakamoto.

Explanation of the Monty Hall problem

Now let me give you my very simple explanation of the Monty Hall problem.

The host offers you an opportunity to reverse your odds.

Imagine an information space that is divided into two partitions. You are placed in one of the partitions but not in both at the same time. You seek truth, i.e. an answer to a certain problem. You are not given certainty about in which partition the truth is located, but you do know the following: the sum of the probabilities that the truth is located in each partition is one (100%).

Now, the host offers you an opportunity to transport you from the partition you are currently located into the other partition.

Whether or not you should switch sides depends on your knowledge or awareness of both the following: (1) your present situation (the odds that you are right in your current partition); and (2) the host’s situation (what the host knows and what he is offering you).

In the Monty Hall problem, your initial odds are 1/3 right and 2/3 wrong; if you switch upon the revelation made by the host, your odds are reversed to 2/3 right and 1/3 wrong.

First, you must be aware that your current situation is a result of how the game is set up (featuring the initial random choice you must make), which does not place you in a neutral situation that is equally favorable to you. Looked at from the information space, your current situation is unfavorable to you. It is important to note that the information space is abstract and mathematical, and it does not depend on how you have made your first random pick. Regardless of which door you pick, you are placed in the unfavorable partition in the information space, because this is how the game is set up and has nothing to do with your act or intellect. You should not confuse this abstract information space with the actual situation you’re in after you have made the first pick (see below for more discussions about this distinction), because you have absolutely zero knowledge of your actual situation. Your only knowledge at this point (before the host makes any revelation) is abstract and mathematical, and you must act accordingly.

Second, you must understand why the host can change your situation. This is possible because, even though you, as a player, start to pick randomly, the host is all-knowing and is doing a favor for you. His action contains a clue. If you pick up the clue, you are partially importing his knowledge into your decision, thus reversing the odds.

For the favor to become a reality, however, you must have a right estimate of your previous odds, as well as a proper appreciation of the host’s intention.

If you feel you’ve got it at this point, you probably don’t need to read the rest of this section and can skip to the next one. But if you are still unsure or just curious, let me explain in further detail below.

The key is to (1) consider the time sequence of events and identify any new information that has been brought in to update your situation and (2) think reversely and exhaustively of all your possibilities and derive an outcome for every possibility. Once you do, the answer is simple.

In Monty Hall problem, you, as the player, made a choice first. There are only two possible outcomes with your choice; you’re either right or wrong, but it is critical to notice that in this particular setting, those two probabilities are initially not equal. The probability that your initial choice is right is 1/3, while the probability that your initial choice is wrong is 2/3.

Remember, (1) your initial choice was either wrong or right, and there isn’t another possibility, and (2) the host knows whether your initial choice was wrong or right, and his next move is relevant to this specific knowledge, thus transforming the purely theoretical scenario into a history-dependent scenario.

In other words, your fate has been updated, and you need to come out of the purely theoretical universe and decide based on your real updated situation.

Now, think reversely to reveal how your fate is updated exactly. Because there are only two possibilities, either right or wrong, you can analyze your fate exhaustively.

If your initial position (pick) was wrong (meaning that you happen to have picked one of the two goats), then the probability that the remaining door has the car is 100% (because the host just picked the other goat, and his pick is NOT independent of your choice).

That is, if you are initially wrong, then changing now would make you 100% correct.

But if your initial position (pick) was right (meaning that you happen to have picked the only car), then changing your choice would mean that the probability that the remaining door has the car is 0% because you already picked the only car in your initial choice.

That is, if you are initially correct, then changing now would make you 100% wrong.

You see, the matter is very clear. The host has offered an opportunity to reverse your odds precisely. Because the probability of your initial choice being wrong is 2/3, if you change now, given the updating new event (that the host has made a new revelation), the total probability of your turning right is also 2/3. Therefore, you have just reversed the probability in light of the new evidence.

Likewise, because the probability of your initial choice being correct is 1/3, if you change it, your total probability of being correct is also 1/3. Again, therefore, you have just reversed the probability in light of the new evidence. (It shows changing in this particular game does have its own risk, but a rational change improves the overall probability of you being correct.)

In the above, you really don’t need to estimate both probabilities of your being right and wrong because either will give you the right explanation. But you do it anyway because you want to have a higher level of mathematical confirmation when the two probabilities add up to 100%, which they must do.

Lesson: situational awareness

Note that the answer in Monty Hall problem has to do with its specific conditions, including that the initial probability of being right is smaller than that of being wrong and also that the host knows both the truth and your initial choice. A change of any of these conditions may lead to a different conclusion.

The lesson of the Monty Hall problem is NOT that you should always change your mind given new evidence, but you should always (1) have a right estimate of your previous condition, i.e., prior probability, and (2) update your posterior probability given new evidence. And you must consider the new evidence logically and mathematically rather than just follow your ‘gut feeling’.

Now, if you are still not convinced and feel it seems like gambling with equal odds, it’s because you are still only looking at this in the present time as if it were an isolated single point and have forgotten where you were from, and what the Host is doing for you.

Remember, in the Monty Hall problem, you did not come from a world of neutrality (meaning equal odds of being right and wrong). You came from a world where you were more likely to be wrong than to be correct, specifically 2/3 versus 1/3.

And what the Host does now is offer you, with his all-knowing knowledge, an opportunity to precisely reverse your fate.

Therefore, if you have the correct estimate of your previous odds, you’d be a fool not to take the opportunity.

Deeper lesson: be in the right game

But there’s also another high-level hint that can be learned from Monty Hall problem:

Watch the Host, and carefully consider the Host’s moves and motivation!

But the question is, who is the Host in your world? A fundamental reason why our life is complicated is that in the limited timeframe (which is our present life, not eternity), two different hosts are trying to guide and influence our game. One is a trickster, another the patient but quiet Truth teller. The Creator of the universe never intends to trick our minds with reality. The Enemy does.

The universe in which we live is not a mere structure. It is a story in which each actor will have a different ending depending on the sequential choices that the actor has made.

Consider the fact that we are placed in a Bayesian game, except that, if you are in the right game, the righteous Host is not to trick you, but to educate you, to test you, and to ultimately reward you; but if you are in the wrong game, both the rules and the outcome are entirely different because of the evil host of that game never intends to give you anything in the end.

Even deeper lesson: the redemption problem

Now, let’s take the Monty Hall problem to the extreme and see how it transforms into a parable of the redemption problem, which concerns a truth about eternity.

Let’s say there are an infinite number of choices, and among them, only one is right. You make your pick first. The Host then takes out all but one other choice and reveals to you that they are all wrong, leaving only two before you now, one you picked first and the other that remains, both unrevealed.

The host then asks you, “Do you wish to stick with your choice or change it?”

Clearly, knowing your initial odds, you’d be a fool if you don’t switch, because what the Host is offering you is changing your fate from being 100% wrong to being 100% right.

Now, what if the outcome concerns the ultimate fate in eternity?

It is a matter of redemption. In this regard, being ‘right’ means being right with God, which translates to eternal life, while being ‘wrong’ means being wrong with God, which translates to eventual death. And in this ultimate ‘game’, the Host points to the case of everyone who has ever lived on earth and reveals a simple fact: every one of them has died (you may consider the number of cases effectively approaching infinity), except that he allows a question mark to be placed on two cases laid before you, yours own and that of Jesus Christ, the Son of God.

My friends, that’s what God has offered to Adam’s offspring. It is redemption, granted in a switch, that transports you from a position that is certainly unrighteous (wrong) into a new position that is certainly righteous (right).

But why the reluctance among people? It is because people err on the following key issues:

(1) to correctly estimate your pre-existing condition (the prior condition); and

(2) to properly appreciate what the Host has done to update your future condition (the posterior condition).

If one believes his initial choice is definitely the right choice, he’s not going to switch. Even if he just has confidence higher than 50% to be right, or he simply believes that these things are totally unknowable and therefore doesn’t care, he’s not going to switch.

But if one realizes that he is most likely wrong with his prior condition and that the ultimate outcome is important, he will switch.

This imports the full meaning of the words spoken by Jesus Christ:

“For the Son of man has come to save that which was lost.” Matthew 18:11, Luke 19:10.

Because only those who realize that they are lost (losing) may switch (repent) for redemption.

I must admit that the above parable cheapens the real story of the Redemption because it has only cold math in it, not taking into account the real heartfelt story of how Adam lost his way by losing all the odds of being righteous with God (“there is not a righteous man, not even one,” Romans 3:11), and more importantly what price Christ has paid by being the only One who’s righteous and made himself available to be everyone’s choice of redemption.

But at least I think I got the math part right. And I’m very thankful for the Host, not for the math, but for the option He has offered.