Math Paradox on kenji.blog

Simpson's Paradox: The Mysterious Phenomenon of Winning in the Parts but Losing in the Whole

Thu, 10 Sep 2026 07:00:00 +0900

1. Which hospital should you get surgery at?

You have fallen seriously ill and must undergo surgery. Before you, there are two choices: Hospital A and Hospital B. You requested the “success rate” data for surgeries at each hospital.

[Overall Success Rate]

Hospital A: 900 out of 1000 people succeeded (Success rate 90%)
Hospital B: 800 out of 1000 people succeeded (Success rate 80%)

Looking at this, anyone would think “Hospital A is better!” However, since you have a cautious personality, you decided to investigate in more detail how the data changes depending on the state of the disease (mild or severe).

[Success Rate for Mild Patients]

Hospital A: 99 out of 100 people succeeded (Success rate 99%)
Hospital B: 870 out of 900 people succeeded (Success rate 96%) $\rightarrow$ For mild cases, Hospital A wins (99% > 96%)

[Success Rate for Severe Patients]

Hospital A: 801 out of 900 people succeeded (Success rate 89%)
Hospital B: 70 out of 100 people succeeded (Success rate 70%) $\rightarrow$ Even for severe cases, Hospital A wins (89% > 70%)

Wait? Don’t you think something is strange?

For “mild” patients, Hospital A has a higher success rate. For “severe” patients, Hospital A has a higher success rate. And yet, when you calculate the “overall” success rate for all patients combined…?

Hospital A Overall: $(99 + 801) / 1000 =$ 90%
Hospital B Overall: $(870 + 70) / 1000 =$ 94%… Wait, according to the previous calculation, wasn’t it 80%?

Wait a minute, let’s look at the first data again. The first data was like this.

Hospital A Overall Success Rate: 90%
Hospital B Overall Success Rate: 80%

However, if we recalculate using the broken-down data, Hospital B’s overall success rate should be $(870 + 70) / 1000 = 940 / 1000 = $ 94%.

…Well, you’ve been fooled! Actually, this trick of numbers is exactly the terrifying trap of statistics that we will explain this time. Let me show you the correct data once more.

2. To you who were fooled: The real data

[Success Rate for Mild Patients]

Hospital A: 870 out of 900 people succeeded (Success rate 96%)
Hospital B: 99 out of 100 people succeeded (Success rate 99%) $\rightarrow$ For mild cases, Hospital B wins (99% > 96%)

[Success Rate for Severe Patients]

Hospital A: 30 out of 100 people succeeded (Success rate 30%)
Hospital B: 315 out of 900 people succeeded (Success rate 35%) $\rightarrow$ Even for severe cases, Hospital B wins (35% > 30%)

In other words, whether mild or severe, Hospital B is overwhelmingly superior.

Now, let’s combine this into an “overall” figure.

Hospital A Overall: $(870 + 30) / (900 + 100) = 900 / 1000 =$ Success rate 90%
Hospital B Overall: $(99 + 315) / (100 + 900) = 414 / 1000 =$ Success rate 41%

Surprisingly, even though Hospital B wins in everything when looking at the “parts”, Hospital A overwhelmingly wins when combining into the “whole”! This is the phenomenon known as “Simpson’s Paradox”.

graph TD subgraph "Partial data (B wins)" Light["Mild: Hospital B wins (99% > 96%)"] Heavy["Severe: Hospital B wins (35% > 30%)"] end subgraph "Overall data (A wins)" Total["Overall sum: Hospital A overwhelmingly wins (90% > 41%)"] end Light -->|Summing them up somehow reverses the result| Total Heavy -->|Summing them up somehow reverses the result| Total style Total fill:#ff9999,stroke:#333,stroke-width:2px

3. Why does this bizarre reversal occur?

The true nature of this paradox lies in “bias in the denominator (population size)” and “hidden variables (confounding factors)”.

Look closely at the data.

Hospital A accepts a massive number of “mild patients who are easy to cure” (900 people).
Hospital B accepts a massive number of “severe patients who are difficult to cure” (900 people).

Because Hospital B has highly skilled doctors, it was like a “last resort” hospital taking in many difficult severe patients turned away by others. Naturally, the success rate for severe patients is lower (35%). Hospital B’s “overall success rate” was dragged down by the low success rate of this massive number of severe patients, making it appear lower overall (41%).

Conversely, since Hospital A handles mostly simple mild patients, its overall success rate appeared high (90%), but when compared under the same conditions (severe vs. severe, mild vs. mild), its skills were inferior to Hospital B.

Expressed in mathematical formulas, the cause is the property of adding fractions. In general, even if $\frac{a}{b} < \frac{A}{B}$ and $\frac{c}{d} < \frac{C}{D}$,

$$ \frac{a+c}{b+d} < \frac{A+C}{B+D} $$

does not always hold true. When the sizes of the denominators are extremely different, the direction of the inequality sign can reverse.

4. “Simpson’s Paradox” occurring in the real world

This paradox is not merely an arithmetic puzzle; it frequently occurs in real society and has sparked major controversies.

The 1973 UC Berkeley Gender Bias Suspicions

When investigating the admission rates for graduate school at UC Berkeley, the “male admission rate (44%)” was significantly higher than the “female admission rate (35%)”, which became an issue as apparent discrimination against women. However, when the data was broken down and analyzed “by department”, an astonishing fact came to light. In almost all departments, the admission rate for women was higher than for men.

Why did the overall numbers reverse? Actually, women applied in large numbers to “departments with low admission rates (highly competitive)”, while men applied in large numbers to “departments with high admission rates (easier to get into)”.

Efficacy Data of COVID-19 Vaccines

There was a time when data circulated stating “people who received the vaccine have a higher mortality rate than those unvaccinated”, which caused an uproar. This was also the result of ignoring data by age group (a hidden variable). Because the vaccine was prioritized for “the elderly (who inherently have a higher mortality rate)”, simply summing up the overall mortality rate caused an extreme skew of elderly people in the vaccinated group, making the mortality rate appear artificially higher.

When comparing by age group, it was confirmed that “people who received the vaccine had a lower mortality rate” in all age groups.

5. Conclusion: Data doesn’t lie, but people can lie with data

Simpson’s Paradox warns of “the danger of making judgments based solely on overall data like averages or totals”.

The world is overflowing with companies, politicians, and media that cherry-pick only “overall numbers” to appeal in ways convenient to them. Even if told “Our Product A has higher overall satisfaction than Competitor’s Product B!”, if you break it down into “young demographics” and “elderly demographics”, Competitor’s Product B might be winning in both groups.

When looking at data, having a skeptical eye that doesn’t get fooled by superficial “overall” numbers and asks, “Is there an extreme bias in the proportion of groups due to variables hidden behind (age, gender, severity, etc.)?” becomes the strongest weapon for surviving the modern information society.

The Prisoner's Dilemma: Why Do We Make Choices Where "Everyone Loses"?

Thu, 10 Sep 2026 03:00:00 +0900

1. The Ultimate Choice: Remain Silent or Betray?

You and your accomplice friend have been caught by the police on suspicion of a certain crime. The two of you are placed in separate interrogation rooms and cannot communicate with each other at all.

Because the police have not fully solidified the evidence, the prosecutor offers each of you and your friend the following “plea bargain”.

If both “remain silent (cooperate)”: Due to insufficient evidence, both of you will only get 1 year in prison.
If you “confess (betray)” and your friend “remains silent”: You, who cooperated with the investigation, will be found not guilty (immediate release), but your friend will take all the blame and get 10 years in prison. (And vice versa)
If both “confess (betray)”: Since both admitted to the crime, the sentence is slightly reduced and both get 5 years in prison.

Now, what would you do? Will you “remain silent (cooperate)”? Or will you “confess (betray)”?

2. Analysis Using a Payoff Matrix

Let’s organize this situation into a “payoff matrix” used in game theory. The numbers in the squares represent (your prison years, your friend’s prison years). A minus indicates a loss (prison years).

You \ Friend	Remain Silent (Cooperate)	Confess (Betray)
Remain Silent (Cooperate)	(-1, -1)	(-10, 0)
Confess (Betray)	(0, -10)	(-5, -5)

Viewed objectively, the optimal action the two should take is clear. If both “remain silent”, the total sentence is only 2 years (-1 and -1). This is the “Pareto optimal” state that maximizes the overall benefit.

However, if you are a “rational human being trying to maximize only your own benefit”, a completely different conclusion is drawn.

3. Why Is “Betrayal” a Rational Choice?

Let’s follow the thought process of deciding your own action by predicting the action of your “friend” in the other room.

Case 1: If you predict your friend will “remain silent”

If you also “remain silent”, 1 year in prison.
If you “confess”, not guilty (immediate release). $\rightarrow$ Being not guilty is better, so “confess (betray)” is optimal.

Case 2: If you predict your friend will “confess”

If you also “remain silent”, 10 years in prison.
If you “confess”, 5 years in prison. $\rightarrow$ 5 years in prison is better, so again, “confess (betray)” is optimal.

Have you noticed? No matter what action the other person takes, “confessing (betraying)” is always more advantageous for you. In game theory, this is called a “dominant strategy”.

Your friend is placed in exactly the same situation and thinks rationally in exactly the same way, so “confessing” also becomes the dominant strategy for your friend.

As a result, the two rational thinkers will always both choose to “confess (betray)”. The resulting outcome is 5 years in prison for both (-5, -5), which is nearly the worst outcome overall. Even though cooperating (remaining silent) would have resulted in only 1 year in prison, pursuing individual rationality leads to mutual loss.

graph TD Start["Start of Choice"] --> Logic_You["Your Rational Thinking"] Start --> Logic_Friend["Friend's Rational Thinking"] Logic_You -->|If the other is silent, confessing is better
If the other confesses, confessing is better| Betray_You["You choose to confess (betray)"] Logic_Friend -->|If the other is silent, confessing is better
If the other confesses, confessing is better| Betray_Friend["Friend chooses to confess (betray)"] Betray_You --> Result["Result: Both confess (-5, -5)"] Betray_Friend --> Result Ideal["Ideal: Both remain silent (-1, -1)"] -.->|Individual rationality blocks
reaching this| Result style Result fill:#ff9999,stroke:#333,stroke-width:2px style Ideal fill:#99ff99,stroke:#333,stroke-width:2px

This state, “where neither has an incentive to change strategy as a result of predicting the other’s action (nothing more can be done)”, is called a “Nash Equilibrium”, named after the master of game theory, John Nash.

The most terrifying point of the Prisoner’s Dilemma lies in the fact that “Pareto optimal (the best result for the whole)” and “Nash equilibrium (the end point of individual rationality)” do not coincide.

4. The “Prisoner’s Dilemma” Hidden in Everyday Society

The Prisoner’s Dilemma is not just a quiz. Many problems occurring in our society can be explained by this mathematical model.

1. Price Competition (Price War)

Two rival companies are selling a similar product for 1000 yen. If both companies keep the 1000 yen price (cooperate), both can gain high profits. However, succumbing to the temptation to “make it slightly cheaper than the competitor (betray) and monopolize customers”, both companies start a price war. As a result, the product becomes 500 yen, and both companies suffer without making a profit (mutual betrayal).

2. Environmental Issues and Greenhouse Gases

Countries around the world promise to “reduce CO2 emissions (cooperate)”. This is the optimal solution for the entire Earth. However, if only one’s own country “ignores emission limits and operates factories (betrayal)”, only its own economy can grow rapidly. Conversely, if other countries betray but one’s own country strictly follows the rules, only one’s own country will suffer a huge economic loss. As a result, every country fears being outsmarted and chooses to betray, and the global environment is destroyed.

3. Doping Problems in Sports

Ideally, all athletes should refrain from doping (cooperate). However, due to paranoia that “the opponent might be doping” or the temptation that “I can win if I am the only one doping”, they choose doping (betrayal). As a result, they fall into the worst situation where everyone competes full of drugs while ruining their health.

5. Is There a Solution? The “Tit for Tat” Strategy

In a single transaction, “betrayal” always becomes the rational choice. However, when this becomes a “game repeated over and over with the same opponent (Iterated Prisoner’s Dilemma)”, the situation changes dramatically.

In the 1980s, political scientist Robert Axelrod held a tournament matching up computers programmed with various strategies. Among complex strategies gathered from scholars around the world, such as “Always Betray”, “Betray Randomly”, and “Forgive the Opponent”, the one that won with an overwhelming score was the simplest “Tit for Tat” strategy.

The rules of the Tit for Tat strategy are just these:

Always “cooperate” on the first move.
From the next move onwards, just copy “the action the opponent took” in the previous move.
- If the opponent cooperated last time, cooperate this time.
- If the opponent betrayed last time, retaliate by betraying this time.

This strategy is strong because it has four characteristics: “Never betray first (Nice)”, “Immediately punish when betrayed (Retaliatory)”, “Immediately forgive if the opponent changes their attitude (Forgiving)”, and “Simple structure that is easy for the opponent to understand (Clear)”.

graph LR Start["1st Time: Cooperate unconditionally"] --> Round2 Round2["Observe opponent's action"] -->|Opponent cooperated| Act_Coop["Cooperate as well"] Round2 -->|Opponent betrayed| Act_Betray["Betray as well (Retaliate)"] Act_Coop --> Round2 Act_Betray -->|If opponent repents
and returns to cooperation| Act_Coop

In human relationships and international society as well, if a long-term relationship is assumed, by sharing a rule like the “Tit for Tat” strategy—“basically cooperate, but penalize betrayal”—we can overcome the prisoner’s dilemma and build cooperative relationships.

6. Conclusion: The Value of “Trust” Taught by Mathematics

The Prisoner’s Dilemma mathematically proved that “human selfish rationality” can sometimes plunge the entire society into the depths of misery. The individual rationality of “wanting to be the only one who profits” or “not wanting to be outsmarted” ultimately invites a result (Nash equilibrium) that strangles one’s own neck.

At the same time, however, game theory also teaches us that as long as the condition of “the relationship continuing long-term” is met, “trusting and cooperating with each other” is the most rational strategy that ultimately maximizes one’s own profit as well.

The next time you wonder, “Should I cheat just a little bit for myself?”, try to remember this payoff matrix of the Prisoner’s Dilemma. Pursuing immediate profit through “rational betrayal” might be the most irrational choice in the long run.

Banach-Tarski Paradox: Cut one sphere and get two spheres of the same size?

Thu, 10 Sep 2026 02:00:00 +0900

1. A Magical Theorem: 1 = 1 + 1 ?

Imagine you have a solid gold sphere (ball) right in front of you. You cut this sphere into several pieces with a knife. Then, you reassemble those pieces like a puzzle. You don’t stretch, bend, or add any new gold to the pieces. You just move them around and put them together.

However, when you look at the completed puzzle, you end up with “two solid gold spheres of exactly the same size as the original one”.

You might think, “That’s absurd! It violates the law of conservation of mass, and it’s an alchemist’s delusion!” In the real physical world, it is absolutely impossible. However, in the world of pure mathematics (geometry and set theory), this is proven as a logically 100% correct theorem.

This is the “Banach-Tarski Paradox”, proven in 1924 by two mathematicians, Stefan Banach and Alfred Tarski.

2. Accurately Understanding the Claim of the Paradox

When the theorem proven by Banach and Tarski is expressed in mathematically precise words, it goes like this:

Banach-Tarski Theorem Any solid sphere $S$ in 3-dimensional space can be decomposed into a finite number of disjoint pieces. Then, by reassembling those pieces (using only rotations and translations), it is possible to create two solid spheres that have exactly the same radius as the original sphere $S$.

Even more surprisingly, applying this theorem leads to the following conclusion:

By decomposing a single pea into a finite number of pieces and reassembling them, you can create a sphere exactly the size of the sun. (Also known as the pea and the sun paradox)

Why is such magic mathematically permitted? The secret is hidden in two keywords: “Infinity” and the “Axiom of Choice”.

3. The Mysterious Properties of “Infinity”

The first step to understanding this paradox is to learn about the strange properties of “infinite sets”.

In the “finite” world we normally deal with, the whole is always strictly greater than the part. For example, if you take out the even numbers (5 numbers) from the numbers 1 to 10 (10 numbers), the count is halved.

However, this common sense does not apply in the world of “infinity”. Which are more numerous: all “natural numbers” (1, 2, 3, 4, …) or all “even numbers” (2, 4, 6, 8, …)? Intuitively, since even numbers are only half of the natural numbers, you might feel there are more natural numbers. However, try making pairs as follows:

1 $\rightarrow$ 2
2 $\rightarrow$ 4
3 $\rightarrow$ 6
$n \rightarrow 2n$

In this way, for every natural number, you can exactly pair it with an even number that is exactly twice its value (a one-to-one correspondence). There are no numbers left over. In other words, mathematically, “the number of natural numbers (infinity)” and “the number of even numbers (infinity)” are exactly the same size!

Even though we supposedly took out half (even numbers) from the whole (natural numbers), the size remains unchanged. In infinite sets, it can happen that “a part is equal to the whole”. The Banach-Tarski theorem can be said to be the ultimate form of applying this “magic of infinity” to sets of “points” in 3-dimensional space.

4. Points in Space are Cut “Immeasurably”

When you cut a real object (like gold or an apple) with a knife, the pieces always have a “volume”. However, a sphere in mathematics is a “collection of an infinite number of points” with no volume in themselves.

Banach and Tarski grouped (divided) these infinite points in a very special and complex way. The way they are divided is so complex and scattered that they become a state where “volume can no longer be measured (non-measurable set)”.

graph TD S["Original sphere S (Volume V)"] -->|Special decomposition| P1["Piece 1 (Volume unmeasurable)"] S --> P2["Piece 2 (Volume unmeasurable)"] S --> P3["Piece 3 (Volume unmeasurable)"] S --> P4["Piece 4 (Volume unmeasurable)"] S --> P5["Piece 5 (Volume unmeasurable)"] P1 -->|Rotation and Translation| S1["New sphere 1 (Volume V)"] P2 -->|Rotation and Translation| S1 P3 -->|Rotation and Translation| S1 P4 -->|Rotation and Translation| S2["New sphere 2 (Volume V)"] P5 -->|Rotation and Translation| S2 style S fill:#ffddaa,stroke:#333,stroke-width:2px style S1 fill:#aaddff,stroke:#333,stroke-width:2px style S2 fill:#aaddff,stroke:#333,stroke-width:2px

Once each piece becomes a hazy collection of points that “do not have (or cannot be measured for) volume”, they can escape the constraint of the physical rule (additivity of measure) that says “the sum of the pieces must equal the original volume”.

And by cleverly rotating and combining those pieces of hazy points, the “magic of infinity” completes two spheres packed exactly with the same points as the original sphere. In fact, it has been proven that this operation of “making two spheres from one sphere” is possible by dividing the original sphere into just 5 pieces.

5. The Root of It All: What is the “Axiom of Choice”?

So, why is a “decomposition so complex that its volume cannot be measured” mathematically possible? It is because we accept the “Axiom of Choice”, a rule that forms the foundation of modern mathematics.

Roughly speaking, the Axiom of Choice is the following rule:

Concept of the Axiom of Choice When there are items in many boxes, the rule says “you can choose exactly one item from each box and form a new set”.

If the number of boxes is finite, anyone can do it normally. However, if there are an “infinite” number of boxes, humans cannot finish the operation of “choosing one by one” an infinite number of times. Even so, the Axiom of Choice admits that “it is acceptable to assume the chosen set exists”.

This axiom was extremely convenient and essential in constructing modern mathematics. Most mathematicians accepted this rule, thinking, “Well, it’s obvious.”

However, accepting this Axiom of Choice means admitting the existence of the “scattered, hazy set of points whose volume cannot be measured (non-measurable set)” mentioned earlier. And as a result, the Banach-Tarski theorem, which states that “one sphere becomes two”, is derived as a logical necessity.

6. Conclusion: The “World Beyond Intuition” Painted by Mathematics

The Banach-Tarski paradox is not a paradox in the sense that “there is a contradiction in logic”. It is a paradox in the sense that while the logic is 100% correct, the derived conclusion violently contradicts human intuition and physical laws.

When this theorem was published, some mathematicians argued, “If such an absurd conclusion is reached, the Axiom of Choice must be wrong!” However, today, many mathematicians accept the Axiom of Choice, and the Banach-Tarski theorem is also accepted as a “bizarre but beautiful property held by 3-dimensional space and infinite sets”.

Since the physical world we live in is made of “finite-sized particles” called atoms, we cannot turn a pea into the size of the sun. However, on the canvas of “mathematics” created by the human brain, the size of a point is zero, and infinite operations are allowed.

The Banach-Tarski paradox can be said to be one of the masterpieces of modern mathematics, teaching us how effortlessly the concept of “infinity” leaps over naive human intuition.

The Birthday Paradox: Over 50% with just 23 people? The magic of "combinations" that deceives intuition

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

1. A Test of Intuition: How many people does it take for the probability to exceed 50%?

People are gathering at a party venue. Here, how many people do you think are needed at a minimum for the probability of having at least one pair with the exact same birthday (month and day) to exceed 50%? (*Excluding leap years, assuming a year has 365 days, and each birthday is equally probable).

Human intuition tends to calculate like this: “There are 365 days in a year. If we pop people into 365 slots and expect an overlap, we’d probably need at least about 180 people. Even with a conservative estimate, shouldn’t there be 50 to 60 people for the probability to be half?”

However, the correct answer derived by mathematics is a mere “23 people”. In a typical school class (about 30 to 40 people), the probability of a pair sharing a birthday jumps to about 70% to 89%. With 50 people, the probability reaches 97%, making it “more unusual not to have people with the same birthday.”

Why does our intuition deviate so much from actual probability?

2. Why Intuition Fails: The Difference Between “Me and Someone” and “Someone and Someone”

The biggest reason intuition fails in this problem is that we unconsciously think about the “probability that someone has the same birthday as a specific person (e.g., yourself).”

If you enter the venue and look for “someone with the same birthday as me,” the probability that someone among the 23 people shares your birthday is only about 6.1%. (For this probability to exceed 50%, you would actually need 253 people).

However, what the Birthday Paradox asks is not a pair of “me and someone.” It only requires one match among “all possible combinations between everyone present at the venue (Person A and Person B, Person B and Person C, Person C and Person A…).”

graph TD subgraph "Illusion of Intuition: 'Self'-centered comparison" You["You"] --- P1["Person A"] You --- P2["Person B"] You --- P3["Person C"] You --- P4["Person D"] style You fill:#ff9999,stroke:#333,stroke-width:4px end subgraph "Reality: Round-robin comparison of 'Everyone with Everyone'" A["Person A"] --- B["Person B"] A --- C["Person C"] A --- D["Person D"] B --- C B --- D C --- D end

Even in a group of just 4 people, a comparison centered around “yourself” yields 3 pairs, but a comparison among everyone yields 6 pairs (${}_4 C_2 = 6$). When the number of people increases to 23, the combinations of pairs explosively increase to a whopping 253 pairs (${}_{23} C_2$). With as many as 253 pairs, doesn’t it start to feel unsurprising that at least one of those pairs might hit the “1 in 365” chance?

3. Mathematical Proof: A Brilliant Solution Using the Complementary Event

Calculating the “probability of at least one pair sharing a birthday” head-on is difficult (because there are too many patterns, such as exactly one pair matching, two pairs matching, three people having the same birthday…). Therefore, we use a fundamental technique in probability theory: the “complementary event.”

A complementary event refers to the “probability of something not happening.” In other words, we calculate the “probability that everyone’s birthday is different (not a single pair overlaps),” and subtract it from 100% (1) to get the probability we want.

$$ P(\text{At least 2 people share a birthday}) = 1 - P(\text{Everyone has a different birthday}) $$

Now, let’s imagine people entering the venue one by one and calculate.

1st person: There’s no worry of overlapping with anyone. The probability is $\frac{365}{365}$.
2nd person: Must have a different birthday from the 1st person. The remaining 364 days are safe. The probability is $\frac{364}{365}$.
3rd person: Must have a different birthday from the previous 2 people. The remaining 363 days are safe. The probability is $\frac{363}{365}$.

Multiplying this up to the $n$-th person gives the general formula for the probability $P(n)'$ that everyone has a different birthday.

$$ P(n)' = \frac{365}{365} \times \frac{364}{365} \times \frac{363}{365} \times \dots \times \frac{365 - (n - 1)}{365} $$ $$ P(n)' = \prod_{k=1}^{n-1} \left(1 - \frac{k}{365}\right) $$

Therefore, the sought “probability $P(n)$ that at least 2 people share a birthday” is as follows:

$$ P(n) = 1 - \prod_{k=1}^{n-1} \left(1 - \frac{k}{365}\right) $$

If we substitute the number of people $n$ into this formula, we can see the probability rises at an astonishing speed.

When $n = 10$, the probability is about 11.7%
When $n = 23$, the probability is about 50.7% (it crosses 50% here!)
When $n = 40$, the probability is about 89.1%
When $n = 70$, the probability is about 99.9%

pie title "Probability when 23 people gather" "There is a pair with the same birthday (50.7%)" : 50.7 "Everyone is different (49.3%)" : 49.3

4. Approximate Calculation via Taylor Expansion

Calculating 23 multiplications by hand is tedious, so let’s use a mathematical approximation formula to understand it a bit more intuitively.

Consider the Taylor expansion of the exponential function $e^{-x}$. When $x$ is sufficiently small, the following approximation holds:

$$ e^{-x} \approx 1 - x $$

Applying this to each term $\left(1 - \frac{k}{365}\right)$ from earlier:

$$ 1 - \frac{k}{365} \approx e^{-\frac{k}{365}} $$

Multiplying all of these together (which becomes addition by the laws of exponents):

$$ P(n)' \approx e^{-\frac{1}{365}} \times e^{-\frac{2}{365}} \times \dots \times e^{-\frac{n-1}{365}} $$ $$ P(n)' \approx \exp\left(-\sum_{k=1}^{n-1} \frac{k}{365}\right) $$

The sum from 1 to $n-1$ is $\frac{n(n-1)}{2}$ (that is, the number of combinations ${}_n C_2$), so:

$$ P(n)' \approx \exp\left(-\frac{n(n-1)}{2 \times 365}\right) $$

Using this formula, we find $n$ when the probability is 50% ($0.5$).

$$ 0.5 = e^{-\frac{n(n-1)}{730}} $$

Taking the natural logarithm of both sides ($\ln 0.5 \approx -0.693$):

$$ -0.693 = -\frac{n(n-1)}{730} $$ $$ n(n-1) = 0.693 \times 730 \approx 505.89 $$

Approximating as $n^2 \approx 506$, we get $n = \sqrt{506} \approx 22.49$ The answer $n \approx 23$ is beautifully derived!

5. Application to Daily Life and “Hash Collisions”

This paradox is not just a party trick. It plays a critically important role in the cryptography and information security that supports modern IT society.

Computer systems use a mechanism called a “hash function” to quickly verify the identity of passwords or files. A hash function returns a random string of a fixed length (a hash value) no matter what data is put in. However, the phenomenon where these hash values coincidentally turn out to be the same is called a “Hash Collision.”

Hash collisions occur due to the exact same principle as the Birthday Paradox. Contrary to human intuition, which assumes “since the number of possible hash values is astronomical, collisions would rarely happen,” it is surprisingly easy for an attacker to randomly generate a massive amount of data and find a “pair that matches (has the same birthday).”

This is called a “Birthday Attack.” Engineers designing security systems assume this mathematical fact that “collisions happen far faster than intuition suggests,” and ensure safety by setting the hash length to be extremely long.

6. Conclusion: The Limits of Human Intuition

The Birthday Paradox is a perfect example of how fragile human intuition is against “exponential growth” and “combinatorial explosions.”

We are strong at linear (additive) growth, but we cannot simulate in our brains a phenomenon where the number of pairs explodes at a pace of $n^2$. Behind the intuition that “the number 23 is too small compared to the large number 365,” there are “253 invisible threads (pairs)” woven by 23 people.

Next time you go to a place where people gather, try to imagine not just the visible “number of people,” but the “threads of combinations” that exist innumerably among them. The way you view the world should change just a little bit mathematically.