Mathematical Paradox on kenji.blog

Can it be filled with paint, but not painted on the surface?: Gabriel's Horn

Thu, 10 Sep 2026 21:00:00 +0900

What would happen if there was a container with a “finite volume, yet an infinite surface area”? Intuitively, it seems impossible, but in the world of mathematics, such a solid figure certainly exists. It is the figure known as “Gabriel’s Horn”, also known as “Torricelli’s Trumpet”.

Discovered in 1641 by the Italian mathematician Evangelista Torricelli, this figure shocked mathematicians and philosophers of the time, sparking fierce debate about the nature of “infinity”.

The Painter’s Paradox

If we liken the properties of this figure to everyday “paint”, the following bizarre paradox occurs.

When filling the horn with paint: Since the volume of the horn is finite (exactly $\pi$), you can completely fill the inside of the horn by pouring just $\pi$ liters (about 3.14 liters) of paint.
When painting the surface of the horn: The surface area of the horn is infinite. Therefore, if you try to paint the inside (or outside) surface of the horn with a brush, no matter how much paint you prepare, you will never finish painting it for eternity.

“You can fill the inside with 3.14 liters of paint, but you need an infinite amount of paint to paint the surface.” Why does this counter-intuitive situation occur?

graph TD A["Gabriel's Horn"] --> B["Calculation of Volume (Integration)"] A --> C["Calculation of Surface Area (Integration)"] B --> B1["Volume = π (Finite)"] B1 --> B2["Can fill the inside with paint"] C --> C1["Surface Area = ∞ (Infinite)"] C1 --> C2["Cannot completely paint the surface"] B2 --> D{"Paradox!"} C2 --> D style A fill:#FFD54F,stroke:#333,stroke-width:2px style B1 fill:#81C784,stroke:#333 style C1 fill:#E57373,stroke:#333,color:#fff style D fill:#F44336,stroke:#333,color:#fff,stroke-width:3px

Mathematical Proof: The Magic of Calculus

Gabriel’s Horn is created by rotating the graph of the function $y = \frac{1}{x}$ (where $x \ge 1$) around the $x$-axis. Let’s calculate the volume $V$ and surface area $A$ of this solid figure using calculus.

1. Calculation of Volume (Why it becomes finite)

The volume $V$ of a solid of revolution is found by integrating the cross-sectional area (a circle with radius $\frac{1}{x}$).

$$ V = \pi \int_{1}^{\infty} \left( \frac{1}{x} \right)^2 dx = \pi \int_{1}^{\infty} \frac{1}{x^2} dx $$

Calculating this definite integral:

$$ V = \pi \left[ -\frac{1}{x} \right]_{1}^{\infty} = \pi (0 - (-1)) = \pi $$

The result converges to a finite value $\pi$.

2. Calculation of Surface Area (Why it becomes infinite)

On the other hand, the calculation for the surface area $A$ is as follows.

$$ A = 2\pi \int_{1}^{\infty} y \sqrt{1 + \left(\frac{dy}{dx}\right)^2} dx $$

Since

$$ \frac{dy}{dx} = -\frac{1}{x^2} $$

, the content inside the square root becomes $1 + \frac{1}{x^4}$. Here, since $\sqrt{1 + \frac{1}{x^4}} > 1$ for all $x \ge 1$, the following inequality holds.

$$ A > 2\pi \int_{1}^{\infty} \frac{1}{x} \cdot 1 dx = 2\pi \left[ \ln x \right]_{1}^{\infty} $$

The natural logarithm $\ln x$ diverges to infinity as $x \to \infty$. Therefore, the surface area $A$, which is larger than that, naturally also diverges to infinity.

The “Trick” Behind This Paradox

Even if it can be proven mathematically correct, it might not make sense to our real-world intuition. “If it can be filled with paint, that paint is touching the inner surface, so shouldn’t the surface be painted as well?”

This discrepancy in intuition arises from the confusion of mathematical concepts with physical reality.

In the world of mathematics, the “thickness” of paint can be made infinitely thin down to zero. Although Gabriel’s Horn becomes infinitely narrow as it goes further, the mathematical paint can become infinitely thin and flow into the deepest depths of that narrow tip, coating the infinite surface area with a finite volume (however, the thickness of the paint film approaches zero towards the tip).

However, in the physical real world, paint is made up of atoms and molecules (particles with a finite size). Even if you pour real paint, once the horn’s tube becomes narrower than the “diameter of a paint molecule”, the paint can go no further. In other words, physically, it is impossible to fill it to the tip, nor to paint its infinite surface.

Gabriel’s Horn is a beautiful example that teaches us that human intuition is bound by the “rules of a finite world” and does not always align with the world of calculus, which deals with “infinity”.

Why did building a new road somehow make traffic worse?: Braess's Paradox

Thu, 10 Sep 2026 21:00:00 +0900

Morning rush hour. You are frustrated by the roads that are jammed every day, but good news has arrived. “To eliminate traffic congestion, the city planning department has built the latest shortcut road!” Everyone must have hoped that they could sleep a little longer starting tomorrow.

However, the next day, when the new road opened, instead of making things better, it caused a much worse traffic jam than before, and everyone’s commute time became longer.

This is not an urban legend or an administrative failure. It is a famous phenomenon in network theory called “Braess’s Paradox”, which was mathematically proven by the German mathematician Dietrich Braess in 1968.

The Paradox Model: 4,000 Commuters

Let’s check with a simple mathematical model why the phenomenon of “everyone slowing down even though the number of roads has increased” occurs.

There are 4,000 drivers heading from the starting point (residential area) to the goal point (business district). Initially, there were only two routes (upper route and lower route) as follows.

Upper Route: Go through a narrow road $A$, and then through a wide highway $B$.
Lower Route: Go through a wide highway $C$, and then through a narrow road $D$.

The “narrow road” gets congested as the number of cars increases, so the travel time takes “the number of running cars $\div 100$” minutes. The “wide highway” never gets congested no matter how many cars come, and always takes “45 minutes”.

Travel Time [Before Road Construction]

The drivers are smart, so they try to choose a faster route even slightly. As a result, the 4,000 people are evenly divided into the upper route (2,000 people) and the lower route (2,000 people).

Upper Route Travel Time: $\frac{2000}{100}$ minutes (narrow road) + $45$ minutes (highway) = $65$ minutes
Lower Route Travel Time: $45$ minutes (highway) + $\frac{2000}{100}$ minutes (narrow road) = $65$ minutes

No matter which route is chosen, the travel time stabilizes at “65 minutes” for everyone.

The Trap of the Shortcut Road

Now, suppose the mayor has built a “dream ultra-high-speed bypass that allows you to travel from Waypoint 1 to Waypoint 2 in 0 minutes (instantly)”.

The drivers have gained a new route option. A driver standing at the starting point thinks like this. “It’s better to use the narrow road A than to use the highway C (45 minutes). Because even at worst, if all 4,000 people chose A, it would only take 40 minutes (4000/100).”

Therefore, all 4,000 people head to “Narrow Road A”. When they arrive at Waypoint 1, they think again. “It’s better to go through the new bypass (0 minutes) and use the narrow road D than to use the highway B (45 minutes). Because even if everyone goes through D, it’s 40 minutes at worst.”

Therefore, all 4,000 people head to “Narrow Road D” through the “New Bypass”.

Travel Time [After Road Construction]

As a result of everyone making “the fastest (rational) choice for themselves”, everyone ends up taking the same route (A → New Bypass → D).

Let’s calculate the travel time.

Narrow Road $A$: $\frac{4000}{100} = 40$ minutes
New Bypass: $0$ minutes
Narrow Road $D$: $\frac{4000}{100} = 40$ minutes
Total: $80$ minutes

Surprisingly, despite the creation of a convenient new shortcut, everyone’s commute time worsened from “65 minutes” to “80 minutes”.

You might think, “Why doesn’t at least one person use the back road (the old route)?”, but if one person chooses the old highway route (45 minutes + 40 minutes = 85 minutes), it will be even slower than the current 80 minutes, so no one tries to change their route. In game theory, this is said to have reached a “Nash Equilibrium”. As a result of everyone taking the optimal action for themselves, it has fallen into the worst outcome as a whole.

Real-world Examples

Braess’s Paradox is not just an armchair theory; it has been observed multiple times in real-world urban traffic and network systems.

1969 Stuttgart, Germany: A new road was built to relieve traffic congestion, but the congestion worsened. Eventually, when the new road was closed off, traffic flow improved.
1990 New York: When “42nd Street”, a mecca of traffic congestion, was completely closed off for an Earth Day event, contrary to the expectations of traffic experts, the overall congestion in Manhattan was dramatically relieved.
Communication Networks: The same phenomenon can occur in internet routing and power grids. The moment a new cable or line is added, data packets can concentrate on the “perceived optimal shortest path”, sometimes causing the entire network to go down.

Braess’s Paradox beautifully illustrates the dilemma of complex societies: “a collection of rational individual choices (egoism)” does not necessarily lead to “an optimal outcome for the whole”. Sometimes, “taking away choices (freedom)” can be to the benefit of everyone.

Berry Paradox: The Contradiction That Occurs When Trying to Define a "Number" Using "Words"

Thu, 10 Sep 2026 11:00:00 +0900

1. Expressing Numbers with Words

We routinely express numbers not only using “Arabic numerals (1, 2, 3…)” but also using “words (Japanese, English, etc.)”.

For example, the number “$10$” can be expressed in various words as follows:

“じゅう” (jū: 3 characters)
“ごの2ばい” (twice five: 5 characters)
“ひゃくの10ぶんの1” (one tenth of a hundred: 9 characters)

In this way, let’s consider explaining a certain number using “Japanese characters”. We will set a limit on the number of characters we can use. Here, let’s consider numbers that can be expressed in Japanese using “19 characters or less”.

Naturally, there is a limit to the numbers that can be expressed in 19 characters or less. The number of Japanese character types (hiragana, katakana, kanji, etc.) is finite, and the number of combinations arranging them in 19 characters or less is also finite (it will be an astronomical number, but it is not infinite).

In other words, there absolutely must exist “huge integers that simply cannot be fully expressed in Japanese using 19 characters or less”.

2. Birth of the Paradox

Now, here is the main point. There are countless “integers that cannot be expressed in Japanese using 19 characters or less”. Suppose we find the “smallest one (the least integer)” among those countless unexpressible numbers.

Let’s call that number $X$. Since $X$ is by definition the smallest among the “numbers that cannot be expressed in Japanese using 19 characters or less”, we can refer to it as follows:

“じゅうきゅうもじいないであらわせないさいしょうのせいすう” (the smallest integer not expressible in nineteen characters or less)

Let’s count the number of characters. “じゅ・う・きゅ・う・も・じ・い・な・い・で・あ・ら・わ・せ・な・い・さ・い・しょ・う・の・せ・い・す・う” …Wait? Even without the punctuation, there are 25 characters. This exceeds “19 characters”.

So, let’s tweak the expression a bit and use kanji to make it shorter.

「十九文字以内で表せない最小の整数」

Now, please count the number of characters in this Japanese phrase.

Surprisingly, it is only “16 characters”.

Something strange has happened. We have just expressed the number $X$ using the “16 Japanese characters” in the phrase “十九文字以内で表せない最小の整数”!

graph TD Define["Definition:
X = The smallest integer not expressible in 19 characters or less"] --> CheckLength{"What is the character count of
『十九文字以内で表せない最小の整数』?"} CheckLength -->|It is 16 characters| Contradiction["Contradiction!
X was expressed in 『16 characters』!"] Contradiction --> Paradox["X is supposed to be 『not expressible in 19 characters or less』
but it is 『expressible in 19 characters or less (16 characters)』"] style Contradiction fill:#ff9999,stroke:#333 style Paradox fill:#ff4444,color:#fff,stroke:#333,stroke-width:2px

Even though $X$ is supposed to be a number that “cannot be expressed in 19 characters or less”, the very words defining it perfectly express $X$ in “16 characters (which is 19 characters or less)”. This is the “Berry Paradox”.

3. Who Created This Paradox?

This paradox was devised in 1904 by a person named G. G. Berry, a librarian at Oxford University. It spread worldwide after the genius mathematician and philosopher representing the 20th century, Bertrand Russell, introduced it in his own paper.

(*In the original English paper, the expression “The least integer not nameable in fewer than nineteen syllables” was used, and the paradox was constructed to work with the number of syllables in English.)

4. Why Did the Contradiction Occur?

The fundamental cause of this paradox lies in the ambiguity and self-reference of the “natural languages (such as Japanese or English)” that we usually use.

Natural Language Cannot Withstand the Rigor of Mathematics

In the world of mathematics, “defining a number” is a highly rigorous process (using equations and symbols). However, in the Berry Paradox, an attempt was made to define a mathematical object (an integer) using the everyday language of humans, such as “expressible” or “not expressible”.

Everyday language is incredibly powerful and flexible, but due to that flexibility, it allows for acrobatic feats like “referring to its own character count”. As a result, it caused a self-contradiction (a paradox of self-reference) where “the definition itself breaks the rules of the definition.”

What Does It Mean to Be “Nameable”?

Furthermore, the definition of the phrase “expressible in 16 characters” is also ambiguous. The phrase “the smallest integer not expressible in 19 characters or less” does not directly point to a specific, concrete number (like $987654321...$, for example). It merely describes indirectly that “there must be a number satisfying the condition.”

Mathematically, a clear distinction must be made between “expressing in a directly calculable form” and “stating indirect conditions in words.” The logical trick is hidden in the fact that these two are conflated to claim, “It could be expressed in 16 characters!”

5. Summary and Impact on the Modern Era

At first glance, the Berry Paradox seems like a mere “wordplay” or “riddle”. However, this problem served as a catalyst that made 20th-century mathematicians deeply recognize the “danger of building the foundations of mathematics using everyday language”.

“We must not define numbers with words. Mathematics must be constructed entirely and exclusively with independent, rigorous symbols.”

This paradox became an important milestone leading to cutting-edge studies that changed the history of mathematics, such as “Gödel’s incompleteness theorems” (there are truths in mathematics that can never be proven) and “Kolmogorov complexity” (the theory of how short information can be compressed) in computer science.

Just 16 characters of Japanese exposed the limits of mathematics. That is the beauty of the Berry Paradox.

St. Petersburg Paradox: How Much Would You Pay for a Gamble with "Infinite" Expected Value?

Thu, 10 Sep 2026 05:00:00 +0900

1. The Dream Gamble with “Infinite” Expected Value

As you walk through a casino, a dealer invites you to play a new coin toss game.

[Game Rules]

You pay an entry fee to start the game.
You toss a coin. If it lands on Heads, your prize money doubles, and you get to toss again.
The game ends as soon as it lands on Tails. You receive the prize money accumulated up to that point.

The initial prize starts at $2.

If it lands on Tails on the 1st toss, you get $2 and the game ends.
If it’s Heads on the 1st and Tails on the 2nd, you get $4 and the game ends.
If it’s Heads on the 1st and 2nd, and Tails on the 3rd, you get $8 and the game ends.
…From then on, as long as Heads keep appearing, the prize doubles to $16, $32, $64… and so on.

graph TD Start["Game Start"] --> Toss1{"1st Coin Toss"} Toss1 -->|Tails (1/2)| End1["End: Win $2"] Toss1 -->|Heads (1/2)| Toss2{"2nd Coin Toss"} Toss2 -->|Tails (1/2)| End2["End: Win $4"] Toss2 -->|Heads (1/2)| Toss3{"3rd Coin Toss"} Toss3 -->|Tails (1/2)| End3["End: Win $8"] Toss3 -->|Heads (1/2)| Toss4{"..."} Toss4 -.->|The longer the streak| Infinite["Prize doubles infinitely!"]

Now, here is a question for you. If the entry fee for this game were “$10,000 (about 1.5 million yen)”, would you participate?

Most people would probably say “I won’t participate”. Because there is a 50% chance of getting Tails on the first toss, meaning you would only get $2 and suffer a massive loss.

However, if you calculate this strictly according to mathematical probability theory (expected value), a surprising fact emerges. Mathematically, whether the entry fee is $10,000 or $100 million, you should participate in this game even if you have to borrow your entire net worth.

Why on earth is that?

2. Let’s Calculate the Expected Value

To determine whether a gamble is “profitable or not”, we use a mathematical metric called “expected value”. The expected value is a number that represents “how much you will make on average per game if you repeat the game many times”. The formula is the sum of all “(prize money) × (probability of getting it)”.

Let’s calculate the expected value for this game.

Probability of getting Tails on the 1st toss: $\frac{1}{2}$ Prize is $2$. Contribution to expected value = $2 \times \frac{1}{2} = 1$ dollar
Probability of getting Tails on the 2nd toss: Getting Heads then Tails, so $\frac{1}{2} \times \frac{1}{2} = \frac{1}{4}$ Prize is $4$. Contribution to expected value = $4 \times \frac{1}{4} = 1$ dollar
Probability of getting Tails on the 3rd toss: Getting Heads, Heads, then Tails, so $(\frac{1}{2})^3 = \frac{1}{8}$ Prize is $8$. Contribution to expected value = $8 \times \frac{1}{8} = 1$ dollar
Probability of getting Tails on the $n$-th toss: $(\frac{1}{2})^n$ Prize is $2^n$ dollars. Contribution to expected value = $2^n \times (\frac{1}{2})^n = 1$ dollar

In other words, no matter what toss the game ends on, the expected value for that pattern is always “$1”. Since the game can potentially continue infinitely, adding all these expected values together results in the following:

$$ \text{Total Expected Value} = 1 + 1 + 1 + 1 + \dots = \infty \text{ (Infinity)} $$

The answer derived by mathematics is “the expected value of this game is infinite”. Since the expected value is infinite, no matter how high the entry fee is, theoretically it is an absolutely “profitable gamble”.

This is the “St. Petersburg Paradox”, proposed by Nicolaus Bernoulli in 1713. There is a fierce contradiction between the correct mathematical calculation result (having infinite value) and human realistic sensation (only wanting to pay a few dollars).

3. The Discovery of “Utility” that Resolves the Gap between Mathematics and Humans

The one who solved this paradox was Daniel Bernoulli, a genius mathematician and cousin of Nicolaus. (It got this name because he presented this paper at the Academy of Sciences in St. Petersburg.)

Daniel delved into human psychology. He thought, “Humans do not judge things by the ‘absolute monetary amount’, but by the ‘satisfaction (utility)’ that the money brings.”

This is called the “Law of Diminishing Marginal Utility”.

The value of money decreases depending on the amount you hold

For example, when you are extremely thirsty in a desert, the first glass of water has enough value (satisfaction) that you would “want to drink it even if you had to pay 10,000 yen”. However, as you drink the second and third glasses, the value of a single glass of water rapidly drops. By the 10th glass, you would likely say “I wouldn’t want it even if it were free”.

The same goes for money.

“1 million yen” given to someone with zero savings has immense, life-saving value.
However, “1 million yen” given to Elon Musk, who has a net worth of tens of billions, only holds about as much value (satisfaction) as a 1-yen coin found on the street.

In other words, even if the prize money doubles infinitely like $2 \rightarrow $4 \rightarrow $8 \rightarrow $16…, the “happiness (utility)” a human feels does not increase infinitely in proportion to the amount.

4. Recalculating the Expected Value using “Utility”

Daniel Bernoulli assumed that “the value (utility) of money felt by humans is proportional to the logarithm ($\log$) of the amount”.

Let the amount be $x$, and let’s express the value (utility) felt by humans $u(x)$ as a logarithmic function (here we consider a simple model with a base of 2).

Utility of $2 prize: $\log_2(2) = 1$
Utility of $4 prize: $\log_2(4) = 2$
Utility of $8 prize: $\log_2(8) = 3$
Utility of $2^n$ prize: $\log_2(2^n) = n$

The amount doubles each time, but human “happiness” only increases little by little like 1, 2, 3… Using this “utility”, let’s calculate the expected value (expected utility) again.

$$ \text{Expected Utility} = \sum_{n=1}^{\infty} \left( n \times \left(\frac{1}{2}\right)^n \right) $$ $$ = 1 \cdot \frac{1}{2} + 2 \cdot \frac{1}{4} + 3 \cdot \frac{1}{8} + 4 \cdot \frac{1}{16} + \dots $$

When you calculate the sum of this infinite series, the result does not become “infinite”, but converges to “2”. If we reverse calculate the amount for which the utility is “2”, it becomes $2^2 = 4$ dollars.

In other words, when recalculated by incorporating human psychology (utility), a very common-sense and realistic answer is derived: **“The value of this game is about ‘$4' according to human sensation."** That is precisely why we do not feel like paying $10,000 for this game.

5. Conclusion: The Paradox that Opened the Door to Economics

The St. Petersburg Paradox was a groundbreaking paradox that mathematically proved that the objective number of “monetary amount” and the subjective value of “human satisfaction” do not match.

The concept of “Utility” proposed by Daniel Bernoulli, 200 years later, became the most important foundation of modern microeconomics and financial engineering (such as portfolio theory). The behavior of us buying insurance or diversifying investments can all be explained by this human psychological mechanism of “diminishing marginal utility (the pain of a huge loss is far greater than the joy of a huge gain)”.

A simple calculation problem of gambling ended up deciphering the human mind and triggering the birth of the massive academic discipline of economics.

Monty Hall Problem: The Trap of Probability Theory that Betrays Intuition and its Complete Resolution using Bayesian Inference

Thu, 10 Sep 2026 00:00:00 +0900

1. The Stage is a TV Quiz Show: What Would You Do?

In 1990, the following question was submitted by a reader to the column “Ask Marilyn” in the American news magazine Parade.

You are a contestant on a TV game show. In front of you are 3 doors (A, B, C). Behind one door is a new car (the prize), and behind the remaining two doors are goats (the blanks).

First, you choose Door A.

Then, the host, Monty Hall, who knows what is behind each door, opens Door B, which has a goat.

Monty says to you, “You may now change your choice to Door C if you like. What will you do?”

So, should you change your door?

Intuitively, it seems: “The remaining doors are A and C. Since the new car is completely random between the two, the probability of winning is $\frac{1}{2}$ (50%) for each. So it doesn’t matter whether you change or not.”

However, columnist Marilyn vos Savant (recognized by the Guinness Book of Records as having the highest IQ) replied, “You should change. If you change, your probability of winning doubles.”

This answer caused a sensation across the United States, bringing in a storm of harsh criticism with about 10,000 letters of protest (about 1,000 of which were from scholars with PhDs in mathematics), saying things like “You do not understand the basics of probability” and “That’s female logic.” However, to get straight to the conclusion, Marilyn’s answer was mathematically entirely correct.

2. The Gap Between Intuition and Mathematics: Branching Probabilities in Mermaid

Why does our intuition create the illusion that it is “$\frac{1}{2}$”? First, let’s visualize all the patterns of the game.

graph TD Start["Game Start"] --> CarA["Car is behind Door A (Prob. 1/3)"] Start --> CarB["Car is behind Door B (Prob. 1/3)"] Start --> CarC["Car is behind Door C (Prob. 1/3)"] CarA --> PickA1["You pick Door A"] CarB --> PickA2["You pick Door A"] CarC --> PickA3["You pick Door A"] PickA1 --> HostB_or_C["Host opens B or C"] PickA2 --> HostC["Host must open C"] PickA3 --> HostB["Host must open B"] HostB_or_C --> Stay1["Stay: Lose..."] HostB_or_C --> Switch1["Switch: Win!"] HostC --> Stay2["Stay: Lose..."] HostC --> Switch2["Switch: Win!"] HostB --> Stay3["Stay: Lose..."] HostB --> Switch3["Switch: Win!"] style Switch2 fill:#bbf,stroke:#333,stroke-width:2px style Switch3 fill:#bbf,stroke:#333,stroke-width:2px style Stay1 fill:#f99,stroke:#333,stroke-width:2px

Assuming you chose “Door A”, the following three scenarios occur with equal probability ($\frac{1}{3}$).

Scenario 1 (Car is A): The host opens either B or C, both of which have goats. If you change your door, you lose.
Scenario 2 (Car is B): The host can only open C, which has a goat. If you change your door, you win.
Scenario 3 (Car is C): The host can only open B, which has a goat. If you change your door, you win.

In other words, in 2 out of 3 times (Scenarios 2 and 3), you are in a state where “you will definitely win if you change doors”. Therefore, the win rate when you change doors is $\frac{2}{3}$, which is double the win rate of $\frac{1}{3}$ when you do not change.

3. Strict Proof Using Bayes’ Theorem

To strictly solve this problem mathematically, we use “Bayes’ Theorem” to calculate conditional probabilities.

$$ P(H|E) = \frac{P(E|H) P(H)}{P(E)} $$

Here, we define the events as follows:

$C_A, C_B, C_C$ : The events that the new car is behind doors A, B, and C, respectively. The prior probabilities are $P(C_A) = P(C_B) = P(C_C) = \frac{1}{3}$
Suppose you initially selected Door A.
$M_B$ : The event that the host opens Door B, which has a goat.

What we want to find is “the probability that the new car is behind Door C given that the host opened Door B”, i.e., the posterior probability $P(C_C|M_B)$.

First, let’s consider the probability $P(M_B|C_X)$ that the host opens Door B depending on where the new car is.

When the new car is behind Door A ($C_A$) The host can open either B or C randomly.
$$ P(M_B|C_A) = \frac{1}{2} $$
When the new car is behind Door B ($C_B$) The host cannot open the door with the new car, so the probability of opening B is zero.
$$ P(M_B|C_B) = 0 $$
When the new car is behind Door C ($C_C$) The host cannot open A (which you picked) or C (where the new car is), so they must inevitably open B.
$$ P(M_B|C_C) = 1 $$

Next, we find the total probability $P(M_B)$ that the host opens Door B using the “Law of Total Probability”.

$$ P(M_B) = P(M_B|C_A)P(C_A) + P(M_B|C_B)P(C_B) + P(M_B|C_C)P(C_C) $$ $$ P(M_B) = \left(\frac{1}{2} \times \frac{1}{3}\right) + \left(0 \times \frac{1}{3}\right) + \left(1 \times \frac{1}{3}\right) = \frac{1}{6} + 0 + \frac{1}{3} = \frac{1}{2} $$

Finally, we apply Bayes’ Theorem to calculate the posterior probabilities for Door A and Door C.

Probability that the new car is behind Door A (if you stay):

$$ P(C_A|M_B) = \frac{P(M_B|C_A) P(C_A)}{P(M_B)} = \frac{\frac{1}{2} \times \frac{1}{3}}{\frac{1}{2}} = \frac{1}{3} $$

Probability that the new car is behind Door C (if you switch):

$$ P(C_C|M_B) = \frac{P(M_B|C_C) P(C_C)}{P(M_B)} = \frac{1 \times \frac{1}{3}}{\frac{1}{2}} = \frac{2}{3} $$

The mathematical proof also clearly demonstrates that “changing doors doubles your probability of winning (2/3)”.

4. Cognitive Bias: The Value of Information as “Conditioning”

Why did even many genius mathematicians intuitively get this problem wrong? The answer lies in the “equiprobability bias” and the “failure to update information” built into the human brain.

4.1. Equiprobability Bias

When presented with unknown options, humans have a tendency to unconsciously assign that “the probabilities of the remaining options are always equal.” The moment we see two doors remaining, our brain automatically labels them as “$50\%$ : $50\%$”.

4.2. The Information of the Host’s “Intent”

The biggest reason intuition goes wrong is overlooking the fact that the host’s actions are not random. If the rule was “the host opens a door randomly without knowing where the car is, and it just happened to be a goat” (known as the Monty Fall problem), then the probabilities for Door A and Door C would both be $\frac{1}{2}$.

However, in the actual Monty Hall problem, the host operates under the following strict constraints:

They cannot open the door chosen by the contestant.
They cannot open the door with the new car.

Because of these constraints, the very act of the host “opening Door B” gives us massive information about Door C. It contains the unspoken message, “I couldn’t open Door C (because the new car is there).”

5. Correcting Intuition with an Extreme Example

If you’re still not convinced, try increasing the number of doors to 1,000,000.

You pick Door 1 out of 1,000,000 doors. (Probability of winning is $\frac{1}{1,000,000}$)
The host, who knows everything, opens all 999,998 doors with goats behind them out of the remaining 999,999 doors.
The only doors closed are “Door 1” which you picked, and “Door 777,777” which the host deliberately left closed.

Now, do you change? In this case, unless you believe you pulled off a “one in a million” miracle right at the start, you should change. Realistically, it should be intuitively clear that the probability of the new car being behind “the single door the host absolutely could not open” is $\frac{999,999}{1,000,000}$.

The Monty Hall problem (with 3 doors) is simply a scaled-down phenomenon of this “1,000,000 doors” scenario.

pie title "Effect of Switching Doors (100 Simulations)" "Win by switching (approx. 66.7%)" : 67 "Win by staying (approx. 33.3%)" : 33

6. Conclusion: Life and Business Lessons from Probability Theory

The Monty Hall problem goes beyond a mere quiz and teaches us important lessons.

Intuition is often wrong: The human brain has not evolved to intuitively process complex conditional probabilities. In important decision-making, relying solely on intuition is dangerous.
Update probabilities with new information (Bayesian updating): When situations change and new information (such as which door the host opened) is provided, the key to success is whether you can flexibly update your probabilities and strategies without clinging to existing beliefs.

The small decision to “change your door” just might double the probability of getting a “new car” in your life.