Bayesian probability and the Monty Hall problem

Bayesian Probability and the Monty Hall Problem

On a rare, slow Friday, I found myself musing on the Monty Hall problem. This famous brainteaser is a probability puzzle where you are a contestant trying to guess which of three closed doors hides a car, while the other two hide goats. Naturally, you want the car. With two goats and one car, a random pick gives you a 1/3 chance of winning – a simple probability most people grasp.

However, there’s a twist: the host knows what’s behind each door (this is crucial). After you make your initial choice and inform the host (also crucial), he doesn’t open your chosen door. Instead, he opens one of the other two doors, revealing a goat. Then, the host asks, “Do you wish to stick with your original choice or switch to the remaining closed door?”.

Let’s cut to the chase: you should absolutely switch. If you stick with your first choice, your winning chance remains 1/3. If you switch, your chance of winning becomes 2/3. This has been extensively discussed, but I’ll offer my explanation later. The correct answer itself is not the main point here, as there’s another important lesson from the Monty Hall problem discussed below. Still, I’m confident my explanation is more intuitive than others you might have encountered.

(On a related note, if you’re interested in Blockchain and Bitcoin, consider reading the article applying Bayes’ theorem to the question of Satoshi Nakamoto’s identity. See “Mathematical proof that Craig S. Wright is Satoshi Nakamoto“. If, after reading both this article and the linked one, you don’t see the connection between identifying Satoshi and the Monty Hall problem, you might be missing the core point of this article.)

Musings on Misconceptions

It’s noteworthy that when the Monty Hall problem was first widely published, a vast majority of people got it wrong. Most believed switching didn’t matter. Their reasoning was that after the host reveals a goat, two doors remain – one with a car, one with a goat – making the odds 50/50, so switching offers no advantage. This is often based on intuition, which, unfortunately, is flawed in this case (as explained later).

Interestingly, many academically trained individuals not only got it wrong but reacted strongly against the correct answer. They felt their understanding of abstract mathematical probability, which they considered infallible, was being disrespected by a mere ‘trick’. The comments from numerous PhDs at the time reflect this sentiment.

The error often lies in mistakenly treating each moment as an independent event, isolated in space-time. Educated individuals might be more prone to this error because they are trained for deep, precise, and abstract analysis, often detached from reality in a compartmentalized way. And when they are wrong, they tend to be more entrenched in the error.

Our world, however, is interconnected, constantly updated with new information – some signal, much noise. Independent thinking is vital for filtering noise, but understanding the interconnectedness of situations is equally crucial for correctly interpreting signals and their relevance.

In essence: maintain the independence of your mind from external opinions, but always recognize the dependence of situations on preceding events in our sequential, consequential world. This isn’t just dialectic; it’s fundamental to being a valid observer in the universe. If one loses mental independence or fails to grasp the world’s consequential nature (or vice versa), their thoughts are more likely to contribute noise than signal.

A key insight from my musings is that the Monty Hall problem wasn’t just a clever puzzle; it tested people’s ability to evaluate evidence and determine the probability of truth. We tend to think of probability as a static estimate based on a single snapshot of evidence. We often assume events are independent, allowing our brains to calculate odds by freezing a moment in time. We think linearly; our minds aren’t naturally attuned to non-linear relationships. Evidence for this includes the common difficulty in intuitively grasping the power of compound interest in investments. While most understand the concept once explained, our intuition often doesn’t keep pace with compounding’s force. Yet, compounding is one of the simplest sequential relationships in the world.

Bayes’ Theorem: Understanding Probability in a Connected World

Put simply, we often struggle to estimate probability correctly according to Bayes’ Theorem. This theorem governs probabilities, showing how to evaluate the degree of belief or confidence in a truth within a world of relatedness and continuity by constantly updating the posterior probability. Specifically, Bayes’ Theorem provides a rational framework for updating our beliefs by incorporating all relevant evidence sequentially, respecting dependencies across space and time, rather than assessing everything in a single snapshot.

Our universe isn’t just a structure; it’s a story.

Once we grasp the interconnected, continuous nature of our world, rational conclusions can differ significantly from gut feelings. Gut feelings are valuable, especially without direct evidence, but relying on it can be perilous when overwhelmed by information (much of it masquerading as evidence), leading to a dulled and misled intuition. We should strive for rationality – mental rationality for material matters and spiritual rationality for spiritual ones. Rationality is integral to honesty and integrity.

(For an example of applying Bayes’ theorem in a real-world context, see “Mathematical proof that Craig S. Wright is Satoshi Nakamoto“.)

Explanation of the Monty Hall Problem

Here’s my straightforward explanation: The host offers you an opportunity to reverse your odds.

Imagine an “information space” divided into two partitions. You exist in one partition at a time. You are seeking the truth (the car’s location). You don’t know which partition holds the truth, but you know the probabilities of the truth being in each partition sum to 1 (or 100%). The host then offers to move you from your current partition to the other one.

Should you switch? The decision depends on understanding both:

1. Your current situation (the odds of being right in your initial partition).
2. The host’s situation (what the host knows and the nature of his offer).

In the Monty Hall problem, your initial odds are 1/3 correct and 2/3 incorrect. When the host reveals a goat and offers the switch, switching reverses your odds to 2/3 correct and 1/3 incorrect.

First, recognize that your initial situation results from the game’s setup (requiring an initial random choice), which places you in a position that is not neutral or equally favorable. From the perspective of the information space, your initial position is unfavorable. Importantly, this abstract information space doesn’t depend on which door you initially picked. Regardless of your choice, the game’s structure places you in the unfavorable partition (1/3 chance of being right). Don’t confuse this abstract space with your specific situation after your first pick (more on this distinction later), because you have zero actual knowledge about whether your specific door hides the car at that point. Your only knowledge before the host acts is abstract and mathematical, and you must act based on that.

Second, understand why the host’s action changes your situation. While you pick randomly, the host knows where the car is and acts based on that knowledge, essentially doing you a favor. His action provides a clue. By understanding this clue, you incorporate some of his knowledge into your decision, thereby reversing the odds. However, to benefit from this favor, you need an accurate assessment of your initial odds and a proper understanding of the host’s intention and action.

(If this makes sense, feel free to skip the rest of this section. If you’re still unsure or curious, let’s delve deeper.)

The key involves two steps:

1. Consider the sequence of events and identify how new information updates your situation.
2. Think backward (reversely) and exhaustively analyze all possibilities to determine the outcome for each. Doing so makes the answer clear.

In the problem, you make the first choice. There are two initial possibilities: your choice is right, or it’s wrong. Crucially, in this setup, these initial probabilities are unequal: 1/3 chance of being right, 2/3 chance of being wrong.

Remember: (1) Your initial choice was either right or wrong, with no other possibility. (2) The host knows if you were right or wrong, and his subsequent action (opening a door with a goat) is dependent on this knowledge. This transforms the scenario from purely theoretical to history-dependent.

The key: Your situation has been updated, and you must make your decision based on this new reality, not just the initial abstract probabilities.

Now, let’s think backward to see exactly how your situation is updated. Since there are only two initial possibilities (right or wrong pick), we can analyze them completely:

• Case 1: Your initial pick was WRONG (you picked a goat). This happens with a 2/3 probability. Since the host knows where the car is and must open a door with a goat that you didn’t pick, the only door left for him to open is the other goat door. Therefore, the remaining closed door (the one you didn’t pick initially and the host didn’t open) must have the car. In this scenario (which occurs 2/3 of the time), switching guarantees a win (100% correct).
• Case 2: Your initial pick was RIGHT (you picked the car). This happens with a 1/3 probability. The host can open either of the two remaining doors, as both hide goats. If you switch, you will move from the car to a goat, guaranteeing you lose (0% correct).

The situation is clear: the host’s action offers a precise reversal of your initial odds.

Since the probability of your initial choice being wrong was 2/3, switching yields a 2/3 probability of being right after the host’s revelation.

Conversely, since the probability of your initial choice being right was 1/3, switching yields a 1/3 probability of being right.

Switching improves your overall probability of winning, although there’s still a risk (1/3 chance) of switching away from the car.

(Note: Analyzing only one case (e.g., the probability of winning by switching if you were initially wrong) would be sufficient, but analyzing both confirms the logic, as the probabilities must sum to 100%.)

Lesson: Situational Awareness

The Monty Hall solution depends on the specific conditions: the initial probabilities (1/3 right, 2/3 wrong) and the host’s knowledge and actions. Changing any condition could change the conclusion. The lesson isn’t always to change your mind when presented with new evidence. Rather, it’s crucial to always:

1. Accurately estimate your prior condition (prior probability).
2. Update your belief (posterior probability) logically and mathematically based on new evidence, rather than relying solely on gut feeling.

If you still feel it’s like gambling with equal odds, you might be viewing the situation only at the present moment, forgetting your starting point and the host’s informative action. Remember, you didn’t start from a neutral position (equal odds). You started where you were more likely to be wrong (2/3 probability). The all-knowing host then provides an opportunity to precisely reverse your fate.

Given the above, if you have an accurate understanding of your initial disadvantageous odds, refusing this opportunity would be foolish.

Deeper Lesson: Be in the Right Game

There’s another, higher-level insight from the Monty Hall problem: Pay close attention to the Host – their moves and motivations.

But who is the “Host” in your life? Life is complicated partly because, within our limited lifespan, different “hosts” attempt to guide and influence our choices. One might be a trickster, the other a patient, quiet teller of Truth. The Creator of the universe doesn’t intend to deceive us with reality; the Enemy does. Our universe is not merely a static structure; it’s a story where each individual’s ending depends on the sequence of choices they make.

We are, in a sense, in a Bayesian game. However, if you’re in the “right game,” the righteous Host aims not to trick, but to educate, test, and ultimately reward you. If you’re in the “wrong game,” the rules and outcomes differ drastically, as the malevolent host of that game has no intention of giving you anything lasting.

Even Deeper Lesson: The Redemption Problem

Let’s extrapolate the Monty Hall problem to illustrate a concept concerning eternal truth – the redemption problem.

Imagine an infinite number of choices, only one of which is correct. You make your pick. The Host then eliminates all other choices except one, revealing they were all incorrect. You are left with two unrevealed options: your original pick and the one remaining alternative. The Host asks, “Do you wish to stick with your choice or switch?”.

Knowing your initial odds were infinitesimally small (effectively zero), switching is the only rational choice. The Host is offering to change your fate from near-certain failure (100% wrong) to near-certain success (100% right).

Now, consider this in the context of ultimate eternal fate – redemption. In this context, being “right” means being right with God, leading to eternal life; being “wrong” means being wrong with God, leading to eternal death. In this ultimate “game,” the Host points to the history of humanity, revealing that virtually everyone has died (an effectively infinite sample), leaving a question mark over just two cases: your own life and the life of Jesus Christ, the Son of God.

This, my friends, mirrors the offer God extends to humanity. It’s redemption, offered through a “switch” – moving from a position of certain unrighteousness (wrong) to one of certain righteousness (right) through faith in Christ.

Why the reluctance to accept this offer? People often err in two key areas, paralleling the Monty Hall mistakes:

1. Correctly estimating their pre-existing condition (the prior condition – our inherent state apart from God).
2. Properly appreciating what the Host (God) has done to update their future condition (the posterior condition – the state offered through Christ).

If someone believes their initial “choice” (their own way or self-righteousness) is definitely correct, or even just likely correct (>50% chance), or if they deem such matters unknowable and irrelevant, they won’t switch.

But if one recognizes their likely “wrongness” in their prior state and understands the ultimate importance of the outcome, they will switch.

This illustrates the meaning of Jesus Christ’s words: “For the Son of man has come to save that which was lost” (Matthew 18:11, Luke 19:10).

Only those who realize they are lost (in the losing position) will switch (repent) to accept redemption.

I acknowledge this parable overly simplifies the profound story of Redemption. It focuses on the logic (“math”) and doesn’t fully capture the depth of humanity’s separation from God (“there is not a righteous man, not even one,” Romans 3:11) or the immense price Christ paid as the only righteous One to become the means of redemption for all who choose Him.

But I believe the underlying logic, the “math” of the situation, holds true. And I am deeply thankful to the Host, God Himself – not just for the logic, but for the gracious option, an unreasonably great favor, He has provided.