Mathematical Paradoxes on kenji.blog

Does 'Testing Positive' = 'Sick'? The Base Rate Fallacy

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

Anyone would panic if they received a “positive (abnormal)” result on a health checkup or cancer screening. However, with knowledge of statistics and probability, you might be able to take a deep breath and stay calm. This is because “testing positive on a highly accurate test” does not necessarily mean “there is a high probability of actually having the disease.”

This is a typical cognitive bias called the “Base Rate Fallacy” or “Base Rate Neglect,” where human intuition significantly misjudges probability calculations.

The Terrifying Health Checkup Problem

Imagine the following situation.

In a certain town, there is an unknown disease that infects 1 in 10,000 people (0.01%). To detect this disease, an excellent test kit with “99% accuracy” has been developed. (* 99% accuracy means that if a sick person takes the test, there is a 99% chance it will correctly determine “positive,” and if a healthy person takes the test, there is a 99% chance it will correctly determine “negative.”)

You happen to take this test, and the result is “positive.” Now, what is the actual probability that you are infected with this disease?

Many people intuitively answer, “Since the test accuracy is 99%, the probability that I am sick must also be 99%.” However, the mathematically correct answer is “about 0.98% (less than 1%).”

Why on earth does the actual probability become less than 1% even with 99% accuracy?

Bayes’ Theorem and Visualizing the Whole

The key to solving this problem lies in considering not only the accuracy of the test but also “how rare the disease originally is (base rate / prior probability).” Let’s visualize this counter-intuitive phenomenon using a large population of 1 million people.

Total Population: 1,000,000 people
Actually Sick People (1 in 10,000): 100 people
Healthy People: 999,900 people

We administer the “99% accurate” test to all 1 million of these people.

1. When Actually Sick People (100) Take the Test

Since the accuracy is 99%, those correctly determined as “positive” are: 100 people × 99% = 99 people (True Positives)

2. When Healthy People (999,900) Take the Test

Since the accuracy is 99%, there are people who are incorrectly determined as “positive” with a 1% probability (false positives): 999,900 people × 1% = 9,999 people (False Positives)

graph TD A["Total Population (1,000,000 people)"] --> B["Sick People (100 people)"] A --> C["Healthy People (999,900 people)"] B -->|99% Correct| B1["True Positives (99 people)"] B -->|1% Failure| B2["False Negatives (1 person)"] C -->|99% Correct| C1["True Negatives (989,901 people)"] C -->|1% Failure| C2["False Positives (9,999 people)"] B1 -.-> D{"Total Number of People Told 'Positive': 10,098 people"} C2 -.-> D style A fill:#ECEFF1,stroke:#333 style B fill:#FFCDD2,stroke:#333 style C fill:#C8E6C9,stroke:#333 style B1 fill:#F44336,stroke:#333,color:#fff style C2 fill:#FF9800,stroke:#333,color:#fff style D fill:#FFF9C4,stroke:#333,stroke-width:2px

The Real Probability That You Are Sick

Now, you have been told by the doctor, “You are positive.” This means that you have entered the group “Total Number of People Told ‘Positive’ (10,098 people)” at the bottom right of the diagram.

What is the proportion of “people who are actually sick (True Positives)” within this group?

$$ \text{Probability of actually being sick} = \frac{\text{True Positives}}{\text{Everyone told they are positive}} = \frac{99}{99 + 9,999} = \frac{99}{10,098} \approx 0.0098 $$

The calculated result is about 0.98%. Despite being told “positive,” the probability that you are healthy (False Positive) is overwhelmingly higher (about 99%).

Why Does Intuition Make Mistakes?

This phenomenon is mathematically explained by “Bayes’ Theorem,” which calculates conditional probability, but the human brain is very poor at this calculation.

The reason we make mistakes is that we are distracted by the specific, intense information provided right in front of us (“Your test result is positive! The accuracy is 99%!”), and we ignore the vast, boring statistical background data (“In the first place, only 1 in 10,000 people has this disease (base rate)”).

Because the “rarity of the disease (0.01%)” is much more extreme than the “inaccuracy of the test (1%),” the slight testing errors quickly swallow up the actual number of sick people.

The “Base Rate Fallacy” Hidden in Society

This illusion causes panic and incorrect judgments not only in medical care but in various situations.

Facial Recognition Systems and Terrorists: Even if a facial recognition camera with 99.9% accuracy finds a “terrorist” at an airport, because the base probability of terrorists is extremely low, almost everyone caught will be innocent civilians with similar faces (False Positives).
Traffic Accidents and Elderly Drivers: Even if you feel it’s dangerous seeing news that “XX% of cars that caused accidents were driven by elderly people,” unless you consider the “proportion of elderly people among all drivers on the road in the first place (base rate),” you cannot know if a specific age group is truly more prone to causing accidents.

The “Base Rate Fallacy” teaches us the importance of statistical thinking: especially when we see shocking numbers or individual cases, we should step back and consider “how likely is that to happen within the whole in the first place (base rate).”

Does seeing a blue apple prove 'ravens are black'? : Hempel's Ravens

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

How do scientists prove theories? Usually, they use “induction,” gathering data by observing the world. For example, if you wanted to prove the hypothesis that “all ravens are black,” you would observe ravens around the world and confirm one by one that they are black.

However, in the 1940s, logician Carl Hempel pointed out a strange logical loophole hidden in this commonplace scientific method. This is the paradox of Hempel’s Ravens, which states that “simply seeing a blue apple or a red shoe serves as evidence that ‘ravens are black’.”

Logical Substitution: The Magic of the Contrapositive

To understand Hempel’s argument, we must recall the concept of the “contrapositive” learned in high school mathematics.

In logic, if a proposition “If A, then B” is true, its contrapositive “If not B, then not A” must also be true (this is called logical equivalence).

Hypothesis $H_1$: “All ravens are black (If it is a raven, then it is black)”

Let’s take the contrapositive of this hypothesis $H_1$. It becomes “If it is not black, then it is not a raven.”

Hypothesis $H_2$: “Everything that is not black is not a raven”

According to the rules of logic, $H_1$ and $H_2$ have exactly the same meaning (equivalence). If one is proven, the other is automatically proven as well.

Proving Ravens Without Seeing Ravens

Now, to confirm hypothesis $H_1$ (ravens are black), every time you find a black raven, the certainty (evidence) of the hypothesis gets a little stronger. This is something everyone can agree on.

However, since $H_1$ and $H_2$ mean the same thing, finding evidence for hypothesis $H_2$ (things that are not black are not ravens) should directly serve as evidence for hypothesis $H_1$.

So, what constitutes evidence for $H_2$? You just need to find something that is “not black and not a raven.”

Suppose there is a “blue apple” on the table. It is not black, and it is not a raven. Therefore, it is evidence supporting $H_2$.
There are “red shoes” in the closet. These are also not black and not ravens. They are evidence for $H_2$.
A “white cloud” is floating in the sky. This is also evidence for $H_2$.

Since evidence for $H_2$ holds the same value as evidence for $H_1$, the following bizarre conclusion is logically derived:

“The more you observe blue apples and red shoes in a room, the more the hypothesis ‘all ravens are black’ is proven to be true.”

graph TD A["Proposition H1: All ravens are black"] <-->|Logical equivalence (Contrapositive)| B["Proposition H2: What is not black is not a raven"] C["Observation: Black raven"] -->|Serves as evidence for| A D["Observation: Blue apple"] -->|Serves as evidence for| B D -.->|Therefore, this should also be evidence for?| A style A fill:#4CAF50,stroke:#333,stroke-width:2px,color:#fff style B fill:#4CAF50,stroke:#333,stroke-width:2px,color:#fff style C fill:#2196F3,stroke:#333,color:#fff style D fill:#FF9800,stroke:#333,color:#fff

Why Is It Counterintuitive?

No ornithologist anywhere in the world grows more convinced that “ravens are black” by looking at a blue apple. Although it should be perfectly correct logically, why does our common sense reject this?

Several approaches have been proposed in the fields of philosophy and statistics to address this paradox.

1. The Bayesian Solution (Difference in Information Content)

The most compelling counterargument from the perspective of modern statistics (Bayesian probability) focuses on the difference in the “strength of evidence (information content).”

In the world, there are overwhelmingly more “things that are not black” than “black things,” and astronomically more “things that are not ravens” than “ravens.”

When you see a blue apple, it certainly serves as evidence that “all ravens are black,” but its value as evidence (the increase in probability) is infinitely close to zero. Confirming just one of the countless “non-black things” in the vast universe raises the probability that “ravens are black” by an amount comparable to the effect of removing a single grain of sand from a desert. On the other hand, directly finding one black raven carries an overwhelmingly greater evidentiary value.

In other words, the Bayesian solution is that logically “a blue apple is evidence,” but practically “its evidentiary value is equal to zero and can be ignored.”

2. The Limits of “Indoor Ornithology”

This paradox highlights how the foundation of science known as “induction (deriving general laws from observation)” rests on a fragile premise. Relying solely on logical equivalence would enable “indoor ornithology,” where one could verify any universal law (“all swans are white,” “no aliens are green,” etc.) simply by observing the junk in a room without ever going outside.

Hempel’s Ravens is a fascinating paradox that shows that the words “evidence” and “proof” we unconsciously use cannot be fully captured by the rules of pure symbolic logic alone.

Is a flying arrow at rest?: Zeno's Arrow Paradox

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

An arrow shot from a bow is flying through the sky. This arrow is certainly moving. However, Zeno, a Greek philosopher from the 5th century BC, developed the following terrifying logic.

“The flying arrow is actually at rest.”

This is neither a joke nor sophistry, but the “Arrow Paradox” that mathematicians and philosophers have seriously debated for 2500 years.

Zeno’s Argument

Zeno’s argument starts with the concept of a “moment in time.”

Time is a sequence of “moments.”
If you capture a single moment (a point where the length of time is zero) like a “photograph,” the arrow “is” at a specific point in space.
At that moment, the arrow is merely “occupying” that space and is not moving. (If it were moving, it would require a “duration of time” rather than a “moment.”)
This holds true no matter which moment you pick out.
If the arrow is at rest at every moment in time, when does it move?

graph TD A["Flying arrow"] --> B["Time is a continuous sequence of moments"] B --> C["Moment t1: Arrow is at rest at position A"] B --> D["Moment t2: Arrow is at rest at position B"] B --> E["Moment t3: Arrow is at rest at position C"] C --> F{"At every moment, the arrow is at rest"} D --> F E --> F F --> G["Conclusion: The arrow is not moving!"] style A fill:#2196F3,color:#fff style F fill:#FF9800,color:#fff,stroke-width:2px style G fill:#F44336,color:#fff,stroke-width:3px

Intuition vs. Logic

“Ridiculous. The arrow is actually flying, isn’t it?” This is probably most people’s first reaction. However, it is actually extremely difficult to point out logically where Zeno’s argument is flawed.

In fact, the ancient Greek philosopher Diogenes is said to have simply stood up and walked around the room in response to Zeno, showing him, “Look, it is moving.” However, this does not constitute a refutation of Zeno’s logic. What Zeno is questioning is not “whether it can move,” but rather, “can we logically explain what it means to move without contradiction?”

Resolution by Calculus (An Attempt)

Calculus, invented by Newton and Leibniz in the 17th century, provided a mathematical answer (at least partially) to this paradox.

In calculus, “the velocity at a certain moment (instantaneous velocity)” is defined as follows:

$$ v(t) = \lim_{\Delta t \to 0} \frac{\Delta x}{\Delta t} $$

In other words, velocity is defined as the “limit” where the change in position $\Delta x$ is divided by the change in time $\Delta t$, as $\Delta t$ approaches infinitely close to zero.

The point here is that “instantaneous velocity” is not the distance traveled within a zero-time duration. It is a quantity defined as the “tendency,” or “limit,” of minute changes before and after that time.

Therefore, the answer from the perspective of calculus is as follows:

“Indeed, if you capture a moment of zero length, the arrow has not moved ‘within’ that moment. However, even at that moment, the arrow has the property of ‘instantaneous velocity (a non-zero limit value).’ Being ‘at rest’ means having an ‘instantaneous velocity of zero,’ but since the instantaneous velocity of a flying arrow is not zero, the arrow cannot be said to be ‘at rest.’”

Remaining Philosophical Questions

While calculus provided a practical solution to Zeno’s paradox, it hasn’t completely settled the philosophical debate.

The concept of a “limit” is strictly a mathematical tool (calculation procedure), and it does not rigorously answer fundamental questions such as “what physically is the smallest unit of time (moment),” “what is continuity,” and “what is the essence of motion.”

In modern physics (quantum mechanics), the possibility that minimum units for time and space (Planck time, Planck length) exist is discussed. If time is not “continuous” but “discrete (digital),” Zeno’s paradox might need to be re-evaluated in a completely different context.

Even after 2500 years, Zeno’s arrow continues to ask us, “What does it mean to move?” and “What is time?”

When Does a Heap of Sand Stop Being a Heap? The Sorites Paradox

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

Imagine a fine heap of sand made up of 10,000 grains right in front of you. Anyone would agree that this is a “heap.” Now, let’s remove just one grain of sand. 9,999 grains. Still a heap, right? Remove one more grain. 9,998 grains. Still a heap.

Let’s keep repeating this process. Removing a single grain shouldn’t turn a “heap” into “not a heap.” However, if you keep applying this logic over and over, you’ll eventually be left with just one grain of sand.

Is a single grain of sand a “heap”?

Of course, no one would call a single grain of sand a “heap.” And yet, we never once rejected the premise that “removing one grain still leaves a heap.” The logic must break down somewhere, but at exactly which grain did the heap stop being a heap?

This is the Sorites Paradox (also known as the Paradox of the Heap), attributed to the ancient Greek philosopher Eubulides in the 4th century BCE.

Logical Structure

This paradox can be expressed in the form of a syllogism:

Premise 1: A collection of 10,000 grains of sand is a “heap.” Premise 2: A heap with one grain of sand removed is still a “heap.” Conclusion: Therefore, even a single grain of sand is a “heap.”

Premise 1 and Premise 2 each sound perfectly reasonable on their own. However, repeatedly applying Premise 2 leads to an obviously false conclusion.

Why This Paradox Is Unsolvable

The crux of the Sorites Paradox is that the word “heap” is inherently vague. There is no clear definition (threshold) for how many grains constitute a “heap.” Such concepts are called “vague predicates.”

Our everyday language is full of such vague terms:

“Tall” — how many centimeters qualifies as “tall”? A person who is 180 cm is “tall.” What if you shave off 1 mm? Another 1 mm?
“Rich” — how much wealth makes someone “rich”? 10 billion yen is “rich.” What if you lose 1 yen?
“Bald” — how few hairs qualifies as “bald”? Zero hairs is “bald.” What if one hair grows back?

All of these share exactly the same structure as the Sorites Paradox.

Philosophical Approaches

1. Epistemicism (The Boundary Exists)

This approach claims that “a precise boundary between heap and non-heap actually exists, but humans simply lack the ability to perceive it.” For example, there may be an exact boundary where “5,837 grains is a heap but 5,836 grains is not” — we just cannot know it.

This is logically tidy, but most people feel intuitively uneasy with this position.

2. Fuzzy Logic (Graduated Truth Values)

Classical logic deals in a binary of “true or false,” but fuzzy logic allows values anywhere between 0 and 1.

For example:

10,000 grains of sand → “Heap-ness = 1.0 (completely a heap)”
5,000 grains → “Heap-ness = 0.7”
100 grains → “Heap-ness = 0.1”
1 grain → “Heap-ness = 0.0 (completely not a heap)”

This method is practical, but it does not fully resolve the paradox. It introduces a new kind of vagueness: “What is the difference between a heap-ness of 0.7 and 0.699?”

3. Supervaluationism

This approach considers all conceivable reasonable boundaries for the word “heap” simultaneously. If something is judged a “heap” under every possible boundary, it is “definitely a heap.” If it is judged “not a heap” under every boundary, it is “definitely not a heap.” The area where opinions diverge is deemed “indeterminate.”

Impact on Modern Society

The Sorites Paradox is not merely a word game — it raises serious problems in the real world of law and policy.

Age of majority: At 17 years and 364 days, you are a “child”; at exactly 18 years and 0 days, you are an “adult.” What fundamentally changes in a single day?
Poverty line: If your income falls 1 yen below the threshold, you are “in poverty”; 1 yen above, and you are “not in poverty.”
Environmental regulations: If pollutant emissions exceed the standard by 0.001 mg, it’s illegal. Right at the standard, it’s legal.

Human language and thought are inherently imbued with vagueness, and attempting to carve the world into clear-cut binary categories may be fundamentally flawed. The Sorites Paradox is a paradox that has perplexed philosophers for over 2,400 years, revealing the fundamental limits of human intellect.

Richard's Paradox: The Contradiction Caused by Infinite Decimals and the "Diagonal Argument"

Thu, 10 Sep 2026 12:00:00 +0900

1. A List of Numbers Definable in Words

“Richard’s Paradox”, published by the French mathematician Jules Richard in 1905, is akin to a relative of the “Berry Paradox” introduced earlier. However, this one is more mathematical and contains a deep contradiction that feels like peering into the abyss of infinity.

First, imagine collecting all the “real numbers between 0 and 1 (decimals) that can be completely defined by English sentences”.

For example, numbers like these:

“Zero point five” $\rightarrow$ $0.5$
“One third” $\rightarrow$ $0.333333...$
“The number formed by the decimal digits of pi” $\rightarrow$ $0.14159265...$

The combinations of sentences that can be expressed in English are simply rearrangements of characters found in a dictionary, so we can assign an “order” to them. (For example, arranging them in order of length, and if they have the same length, arranging them in alphabetical order.)

In this way, we have created an infinitely continuing numbered list (1st, 2nd, 3rd…) of “all real numbers definable in English”.

$$ \begin{align*} r_1 &= 0.\mathbf{3}333... \\ r_2 &= 0.5\mathbf{0}00... \\ r_3 &= 0.14\mathbf{1}5... \\ r_4 &= 0.777\mathbf{7}... \\ &\vdots \end{align*} $$

Within this list, “every possible real number definable in English” should be perfectly included without a single exception.

2. The Demonic Technique: “Diagonal Argument”

Here, Richard performs a terrifying operation. He artificially creates a “completely new number $X$” that avoids all the numbers currently in the list.

The method is simple:

Look at the 1st decimal digit of the 1st number in the list (in the above example, $3$). Add $1$ to it, and make that the 1st digit of $X$ ($3+1=4$).
Look at the 2nd decimal digit of the 2nd number in the list (in the above example, $0$). Add $1$ to it, and make that the 2nd digit of $X$ ($0+1=1$).
Look at the 3rd decimal digit of the 3rd number in the list (in the above example, $1$). Add $1$ to it, and make that the 3rd digit of $X$ ($1+1=2$).

*If the original digit is $9$, let’s assume it loops back to $0$.

The new number $X$ created by this method (in the example above, $X = 0.4128...$) will absolutely never match any number in the list. This is because the $n$-th decimal digit of $X$ is intentionally shifted from the $n$-th decimal digit of the $n$-th number. (This technique is called the “Diagonal Argument”, devised by the genius mathematician Cantor to prove the infinite size of real numbers.)

3. The Completion of Richard’s Paradox

Now, here comes the paradox.

We have just created a new number $X$. And the “rule” for creating this $X$ is perfectly explained (defined) by the English sentences I just wrote above.

In other words, $X$ is a “real number definable in English”.

However, remember the initial premise. “Real numbers definable in English” were supposed to be all comprehensively included in the initial list ($r_1, r_2, r_3...$). Yet, $X$ was constructed so that it does not match any number in the list.

$X$ must exist within the list (because it was defined in English).
$X$ must not exist within the list (because it was constructed using the diagonal argument to differ from all numbers in the list).

A perfect contradiction! This is Richard’s Paradox.

4. Why Did the Logic Collapse? (The Trap of Meta-language)

The reason this paradox arose, similarly to the Berry Paradox, lies in the confusion of “levels of language.”

To perform mathematics rigorously, one must clearly separate the “list of numbers in question (object language)” from the “rules that talk about the properties of that list from the outside (meta-language)”.

Richard’s list is a collection of “definitions of computable numbers”. However, the rule to create the new number $X$, “look at the $n$-th digit of the $n$-th number in the list”, is a “meta-language” operation that cannot be executed without looking down at the list itself from the outside.

Richard’s Paradox exploded into self-contradiction because it secretly tried to slip the “meta-linguistic number $X$ created by manipulating the list from the outside” into the “inside list”.

5. Passing the Baton to Gödel

This Richard’s Paradox sent a massive shockwave through the mathematical community of the time. “Human language (and logical systems) can easily cause self-contradiction if we are not careful. How can we make mathematics perfect and free of contradiction?”

In 1931, it was the 25-year-old genius mathematician Kurt Gödel who brought a final resolution to this problem. Gödel perfectly translated and reproduced the structure of this paradox, which Richard caused using the “ambiguity of language,” by using “rigorous mathematical formulas (Gödel numbering)”.

The result derived from this was the famous “Gödel’s Incompleteness Theorems”. It was a monumental discovery proving the limits of human knowledge: “No matter how rigorously mathematical rules are established, ’truths that can neither be proved nor disproved’ will inevitably arise within those rules (mathematics is incomplete).”

Richard’s Paradox began as a mere contradictory play on words, and eventually evolved into the ultimate weapon to shatter the “absoluteness” of mathematics itself.

Sleeping Beauty Paradox: Is the coin probability 1/2 or 1/3? A difficult problem dividing probability theory

Thu, 10 Sep 2026 09:00:00 +0900

1. The Rules of a Strange Experiment

You (Sleeping Beauty) have been selected as a subject for a certain scientific experiment. The experiment runs from Sunday to Wednesday. On Sunday night, you are given a sleeping pill and fall asleep.

After you fall asleep, the experimenter tosses a single fair coin (a coin with an exactly 1/2 probability of landing heads or tails). And depending on the result, you will be awakened according to the following schedule.

[If the coin toss result is “Heads”]

You will be awakened once on Monday and asked a question. After that, you will be put back to sleep and will not awaken again until the experiment ends (Wednesday).

[If the coin toss result is “Tails”]

You will be awakened on Monday and asked a question. After that, you will be given a special drug (an amnesia drug) and put back to sleep.
You will be awakened once more on Tuesday and asked the same question. After that, you will be put back to sleep again, and the experiment will end (Wednesday).

Due to the effects of the amnesia drug, when you awaken, you will not be able to remember “what day of the week it is today” or “whether you have been awakened in the past” at all.

graph TD Sunday["Sunday: Beauty goes to sleep"] --> Toss{"Coin Toss"} Toss -->|Heads (1/2)| Mon_Heads["Monday: Awaken + Question
(Then experiment ends)"] Toss -->|Tails (1/2)| Mon_Tails["Monday: Awaken + Question
(Then amnesia)"] Mon_Tails --> Tue_Tails["Tuesday: Awaken + Question
(Then experiment ends)"] style Toss fill:#ff9999,stroke:#333 style Mon_Heads fill:#aaffaa,stroke:#333 style Mon_Tails fill:#aaffaa,stroke:#333 style Tue_Tails fill:#aaffaa,stroke:#333

Now, it is Monday (or Tuesday), and you have awakened. There are no clocks or calendars in the room, so you do not know what day it is today.

There, the experimenter comes in and asks you this question. “Given that you are currently awake, what do you think is the probability that the tossed coin was ‘Heads’?”

You are a beauty well-versed in mathematics. Now, how do you answer?

2. Two Clashing Factions: 1/2 or 1/3?

This problem was devised in the 1990s and published in an academic journal by philosopher Adam Elga in 2000. The probability of the coin seems obvious, but actually, over this problem, mathematicians, statisticians, and philosophers around the world have split cleanly into two camps: the “1/2 Faction (Halfers)” and the “1/3 Faction (Thirders)”, engaging in fierce debate to this day.

Let’s hear the “perfect logic” of each camp.

The Claim of the “1/2 Faction (Halfers)”

“Since the coin is a fair coin with no cheating, the probability of heads coming up is naturally 1/2. No matter how many times the experimenter wakes me up or erases my memory after I fall asleep, it does not affect the physical outcome of the coin at all. The probability at the time the coin was tossed was 1/2, and my waking up gives me no new information (clues to guess whether it is heads or tails). Therefore, the probability remains 1/2.”

This is a very sound opinion that emphasizes objective physical phenomena and the non-update of information.

The Claim of the “1/3 Faction (Thirders)”

“The very fact that you are ‘awake’ is information that changes the probability. Suppose we repeated this experiment 100 times (100 weeks). The coin should be ‘Heads’ 50 times and ‘Tails’ 50 times.

For the 50 weeks where Heads comes up, you awaken only once on Monday $\rightarrow$ Number of awakenings for ‘Heads’ is 50 times

For the 50 weeks where Tails comes up, you awaken twice, on Monday and Tuesday $\rightarrow$ Number of awakenings for ‘Tails’ is 100 times

In other words, the situation of the moment you wake up has 150 times in total, out of which the ‘pattern of awakening on Heads’ is 50 times, and the ‘pattern of awakening on Tails’ is 100 times. Therefore, the probability that the awakening you are currently experiencing is ‘Heads’ is 50 / 150 = 1/3!”

This is a powerful opinion based on “frequentism” or the “anthropic principle,” which incorporates the very situation that “you currently exist (are observing)” into the calculation as an element of the probability space.

3. Calculating with Bayes’ Theorem

There is also an attempt to unravel this problem using “Bayes’ Theorem,” a tool for mathematically updating probabilities. Let’s organize the logic of the “1/3 faction” from the perspective of conditional probability.

Your state when you awaken is one of the following three:

$E_1$: The coin is “Heads”, and it is now “Monday”
$E_2$: The coin is “Tails”, and it is now “Monday”
$E_3$: The coin is “Tails”, and it is now “Tuesday”

The probability of “Heads” is $1/2$, and the probability of “Tails” is $1/2$. However, in the case of Tails, “Monday” and “Tuesday” are perfectly symmetrical (you cannot distinguish them since you have no memory), so it is thought that $E_2$ and $E_3$ are equally likely to occur.

Since the sum of the overall probabilities must be $1$, if we assign equal probability to each awakening as an independent “event (observation point)”: $P(E_1) = 1/3$ $P(E_2) = 1/3$ $P(E_3) = 1/3$ Thus, the conclusion is that the “probability it was Heads ($P(E_1)$)” is $1/3$.

On the other hand, the “1/2 faction” argues against this, stating, “Monday and Tuesday when the coin is Tails ($E_2$ and $E_3$) are merely dependent events derived from the single result of one coin toss, and it is wrong to count them as independent probabilities in the first place.”

4. Why Does This Problem Not Get Resolved?

The reason why the “Sleeping Beauty problem” plagues scholars so much is not due to a mere calculation error or illusion. It is because this problem touches upon the deepest and most fundamental question of probability theory: “What exactly is probability?”

For the 1/2 faction, probability is a “physical property of the coin” or an “objective fact”.
For the 1/3 faction, probability is the “degree of belief of the observer (Beauty)” or the “frequency of observation”.

Profound themes that connect to the “measurement problem” in quantum mechanics and the “anthropic principle” in cosmology (the idea of calculating the probability of the universe backward from the fact that we exist) are condensed into this simple coin toss experiment.

5. Summary

If you were a subject in this experiment, would you answer “1/2” or “1/3” when you wake up?

Whichever you answer, world-class mathematicians will stand behind you to defend you. How a seemingly simple mathematical definition collapses the moment it is tied to troublesome concepts like human “subjectivity” and “existence.” The paradox continues to shake our common sense today.

Newcomb's Paradox: Can You Beat a Superbeing Who Foresees the Future?

Thu, 10 Sep 2026 08:00:00 +0900

1. The Ultimate Choice Game

An alien with superintelligence who calls himself “Omega” appears before you. Omega is a master of analyzing human behavior and possesses the terrifying ability to “predict what choice a subject will make next with almost 100% accuracy.” In past experiments, Omega’s predictions have never been wrong.

Omega places two boxes in front of you.

Box A: A transparent box. It contains exactly 100,000 yen.
Box B: An opaque box. It contains either 100 million yen or is empty (0 yen).

Omega tells you to choose one of the following two actions.

Choice 1: “Take both boxes”: You get both the 100,000 yen in Box A and the contents of Box B.
Choice 2: “Take only Box B”: You get only the contents of Box B. You must give up the 100,000 yen in Box A.

Hearing just this, anyone would decide to “Take both boxes”. However, Omega adds a terrifying “rule”.

[Omega’s Rule]

“Yesterday, I already predicted ‘which choice you will make’ today and set the contents of Box B. If I predicted you would greedily choose to ‘Take both boxes’, I left Box B empty. If I predicted you would not be greedy and choose to ‘Take only Box B’, I put 100 million yen in Box B.”

Now, you must make a choice. Should you “Take both boxes”? Or should you “Take only Box B”?

graph TD Omega["Omega's Prediction
(already completed yesterday)"] Omega -->|Predicts 'Take both'| BoxB_Empty["Box B is empty (0 yen)"] Omega -->|Predicts 'Take only Box B'| BoxB_100M["Puts 100 million yen in Box B"] You["Your Choice
(today)"] You -->|Choice 1: Take both| Result1["Box A (100k) + Box B contents"] You -->|Choice 2: Take only Box B| Result2["Box A (0 yen) + Box B contents"] BoxB_Empty -.-> Result1 BoxB_100M -.-> Result2

2. Two Clashing “Perfect Logics”

This problem was devised by physicist William Newcomb in 1969 and published by philosopher Robert Nozick. As soon as it was published, the opinions of brilliant mathematicians and philosophers around the world were split right down the middle, causing a massive controversy.

This is because there is an “absolutely irrefutable, perfect logic” for either choice.

Logic 1: The argument of the “Take only Box B” camp (Expected Value Maximization)

“Omega’s prediction accuracy is almost 100%, right? Then we should trust Omega based on past data. If I choose to ‘Take both’, Omega has foreseen it, and the result is just 100,000 yen. If I choose to ‘Take only Box B’, Omega has foreseen it, and the result is 100 million yen. Even a fool knows whether they want 100,000 yen or 100 million yen. Therefore, I should absolutely ‘Take only Box B’!”

This thinking is based on “Expected Utility Theory,” which straightforwardly believes in past statistical data and expected values.

Logic 2: The argument of the “Take both boxes” camp (Dominant Strategy)

“Wait a minute. Omega predicted and put the contents in Box B ‘yesterday’, right? That means, at this moment, the contents of Box B are already determined to be either ‘contains 100 million yen’ or ’empty’, and it will never change.

Pattern 1: If Box B already contains 100 million yen, choosing to ‘Take both’ gives me 100.1 million yen, and choosing ‘Only B’ gives me 100 million yen. Pattern 2: If Box B is already empty, choosing to ‘Take both’ gives me 100,000 yen, and choosing ‘Only B’ gives me 0 yen.

In either pattern, choosing to ‘Take both’ absolutely gets me 100,000 yen more! Whatever I choose now, Omega’s actions yesterday cannot be rewritten by a time machine. Therefore, I should absolutely ‘Take both boxes’!”

This thinking is based on the “Dominant Strategy” in game theory, which says to “choose the option that is advantageous to you regardless of what action the opponent takes.”

3. Do you believe in “Free Will”?

“The camp that takes only Box B” and “The camp that takes both”. After hearing both arguments, which did you think was right?

Actually, to this day, there is no “mathematically perfect single correct answer” to this paradox. Because at the root of this problem lies humanity’s greatest philosophical question: “Determinism vs. Free Will”.

Those who answered “Take only Box B” (Determinists)

People who make this choice subconsciously accept “Determinism (everything in the future of this world is determined from the beginning)”. The fact that Omega can predict the future with 100% accuracy means that your current decision was not chosen by “your free will,” but rather “you were already destined to choose that way since yesterday by the physical laws of the universe and the movement of neurons in your brain.” Since the future cannot be changed, this view holds that it is most rational to follow Omega’s prediction and ride the “destiny of taking only Box B.”

Those who answered “Take both boxes” (Free Will advocates)

People who make this choice subconsciously believe in “Free Will (you can carve out the future through your own choices)”. Because they believe that “regardless of Omega’s prediction yesterday, I can change my choice with my own will right now,” they take the action to “add 100,000 yen at this very moment, regardless of the already determined contents of the box.” Even if the result is that Omega predicted it and the box is empty, they possess the logic to accept that “it cannot be helped because it is the result of taking logically correct action.”

4. Time Travel and the Collapse of Causality

What makes Newcomb’s paradox even more complicated is the reversal of “Causality (cause and effect)”.

In the common-sense world we live in, “My choice today (cause)” creates “Tomorrow’s result”.

However, in Omega’s game, It seems that “My choice today (cause)” determines “Yesterday’s Omega’s action (result)”. A “backward causality” occurs, where future actions determine past facts.

If a “perfect predictor” like Omega exists in the universe, even our common sense that “time flows from the past to the future” collapses.

5. Conclusion: A Thought Experiment that Uncovers Human “Rationality”

Which box will you open?

More than half a century has passed since this paradox was presented, but in surveys of philosophy and economics, opinions are beautifully split half-and-half between the “Take both” camp and the “Take only B” camp. And interestingly, both camps genuinely believe that “the opponent’s logic is completely bankrupt and foolish”.

“What is a rational judgment?” No matter how much economics and mathematics develop, in the end, it comes down to the philosophy of “how humans perceive this world.” Newcomb’s paradox is a magnificently mean and beautiful thought experiment that confronts us with the limits of logic.

Russell's Paradox: Does the "set of all sets that do not contain themselves" contain itself?

Thu, 10 Sep 2026 04:00:00 +0900

1. The “Barber Paradox” That Struck a Peaceful Village

In a certain peaceful village, there was a single barber. At the entrance of the village stood a strange sign with the following rule:

“The barber of this village shaves all and only those villagers who do not shave themselves.”

The villagers were satisfied with this rule. Those who couldn’t shave themselves just had to go to the barber, and those who could shave themselves would just do it at home.

However, one day, the young barber looked in the mirror and was suddenly startled. He had stubble growing on his chin. “Now then, should I shave my own beard?”

He decided to think logically according to the rule on the sign.

What if he decides to “shave himself”? According to the rule, the barber must only shave those who “do not shave themselves.” Therefore, if he shaves himself, he is not eligible to be shaved by the barber (himself). In other words, “he must not shave himself.”
What if he decides “not to shave himself”? According to the rule, the barber must shave everyone who “does not shave themselves.” Therefore, if he doesn’t shave himself, he must be shaved by the barber (himself). In other words, “he must shave himself.”

“If he shaves himself, he must not shave himself.” “If he does not shave himself, he must shave himself.”

The barber completely panicked and became unable to take either action. This is the famous “Barber Paradox”.

graph TD Barber["Barber: Should he shave himself?"] Barber -->|YES: Shaves himself| Cond1["Rule violation!
(He must not shave the beard of someone who shaves himself)"] Barber -->|NO: Does not shave himself| Cond2["Rule violation!
(He must shave the beard of someone who does not shave himself)"] Cond1 --> Paradox["Contradiction (Paradox)"] Cond2 --> Paradox style Paradox fill:#ff4444,color:#fff,stroke:#333,stroke-width:2px

2. “Russell’s Paradox” That Shook the Mathematical World

This “Barber Paradox” is an allegory created by the British logician and philosopher Bertrand Russell to explain a mathematical paradox he discovered in a way that is easy for the general public to understand.

What he actually discovered was not about a barber, but a terrifying contradiction regarding “Sets”. It is called “Russell’s Paradox (1901)”.

The Concept of a “Set of Sets”

In mathematics, a “set” is a collection of things that satisfy a certain condition.

“The set of even numbers less than or equal to 10” = $\{2, 4, 6, 8, 10\}$
“The set of red apples”

And, as the contents (elements) of a set, you can also put in another “set”. For example, consider “the set of all books in the world”. Since this set itself is not a “book”, “the set of all books in the world” is not included within its own set.

On the other hand, consider “the set of things that are not books”. This set itself is also not a “book”. Therefore, “the set of things that are not books” is included within its own set.

In this way, sets in the world can be broadly divided into two types:

A: Sets that do not contain themselves (e.g., the set of books)
B: Sets that contain themselves (e.g., the set of things that are not books)

The Birth of the Demonic Set $R$

Here, Russell considered the following special set $R$:

Set $R$ = The set of all “sets that do not contain themselves (Type A)”

Written as a mathematical formula (intensional notation), it looks like this:

$$ R = \{ x \mid x \notin x \} $$

Now, here is the main point. Russell posed the following question regarding this set $R$:

“Does the set $R$ contain itself ($R$)?”

Let’s think about it.

What if $R$ “contains itself ($R \in R$)”? The condition to be included in $R$ is “not containing oneself”. Therefore, $R$ does not satisfy the condition, and cannot be included in $R$. (Leading to $R \notin R$, a contradiction)
What if $R$ “does not contain itself ($R \notin R$)”? The condition to be included in $R$ is “not containing oneself”. Therefore, $R$ perfectly satisfies the condition, and must be included in $R$. (Leading to $R \in R$, a contradiction)

Written as a formula, it is a logical collapse in just one line:

$$ R \in R \iff R \notin R $$

“If it contains itself, it is not contained.” “If it is not contained, it contains itself.” This has exactly the same structure as the Barber Paradox. However, while the village barber story can be laughed off with “the mayor who put up a sign with such rules is just stupid”, in the world of mathematics, it’s not that simple.

This was because the mathematical community at the time was right in the middle of trying to rebuild all of mathematics based on the naive rule (Naive Set Theory) that “as long as you clearly define the condition, you can freely create a ‘set’ out of anything”.

3. Frege’s Tragedy

The person to whom Russell sent a letter detailing this was the great German logician Gottlob Frege. Frege had just sent the second volume of his magnum opus, The Basic Laws of Arithmetic, to the printing press, having dedicated his entire life to it. This book was a culmination of his attempt to prove the completeness of mathematics based on the rule that “a set can be created from any condition”.

Upon reading Russell’s letter, Frege despaired. It proved that using the “most fundamental of foundations” rules in his book, an “absolutely contradictory set” like Russell’s Paradox could be created. If the foundation collapses, hundreds of pages of mathematical formulas built upon it all become invalid.

At the very end of his book, right before publication, Frege left the following agonizing postscript:

“Hardly anything more unfortunate can befall a scientific writer than to have one of the foundations of his edifice shaken after the work is finished. This was the position I was placed in by a letter of Mr. Bertrand Russell, just when the printing of this volume was nearing its completion.”

4. Overcoming the Crisis: The Birth of Axiomatic Set Theory

Russell’s Paradox triggered a massive panic in the mathematical community, known as the “foundational crisis of mathematics”. The freewheeling rule that “as long as you decide the condition, you can freely create a set” had given birth to a monster called contradiction.

To resolve this crisis, mathematicians set out to strictly enforce the rules. Mathematicians such as Zermelo and Fraenkel established a rulebook (axiomatic system) that strictly distinguishes between “sets that are allowed to be created” and “sets that must not be created (too large)”. This is called the “ZFC Axioms (Axiomatic Set Theory)”.

Under the ZFC axioms, a “set $R$ that collects all ‘sets that do not contain themselves’” as conceived by Russell was banned from the world of mathematics, deemed “too huge and dangerous, so it is no longer recognized as a ‘set’ (it is merely a ‘class’)”.

graph LR subgraph "Naive Set Theory (Before Russell)" Free["You can freely create a set
with any condition!"] --> Monster["The Monster of Contradiction R
(Russell's Paradox)"] end subgraph "Axiomatic Set Theory (Modern Mathematics)" Strict["Only those that follow strict
rules (axioms) are 'Sets'"] --> Safe["Contradiction R is not recognized
as a 'Set', so it's safe!"] end Monster -.->|Crisis in the Mathematical World| Strict

5. Conclusion: Paradoxes are a “Drastic Medicine” for Fixing “Logic Bugs”

Russell’s Paradox is the ultimate logic bug caused by self-reference (referring to oneself), similar to “a snake eating its own tail (Ouroboros)” or “the liar who says ‘I am a liar’”.

At first glance, paradoxes may seem like mere sophistry or wordplay, but they destroyed the very foundation of mathematics, the most rigorous of disciplines, and consequently forced it to evolve into something stronger and more rigorous.

If the genius Russell had not noticed this “barber’s bug”, modern mathematics, and computer science which lies on the extension of that logic, might have developed while harboring a fatal contradiction somewhere. A paradox is the most stimulating drastic medicine that teaches us the limits of human logic.

Achilles and the Tortoise: Can He Catch Up or Not? The Paradox of "Infinity" from Ancient Greece

Thu, 10 Sep 2026 01:00:00 +0900

1. Zeno’s Paradox: Can the Swift Hero Not Beat the Tortoise?

In the 5th century BC, the ancient Greek philosopher Zeno of Elea presented several paradoxes regarding “motion” that directly contradicted our intuition and common sense. The most famous among them is the paradox of “Achilles and the Tortoise”.

Achilles, the swiftest hero in Greek mythology, and a tortoise, the epitome of slowness, have a footrace. Of course, Achilles is overwhelmingly faster, so the tortoise is given a handicap and allowed to start a little further ahead.

The race begins. Achilles chases after the tortoise at breakneck speed. However, Zeno argued as follows:

“Achilles will never be able to catch up to the tortoise.”

Why on earth is this? Zeno’s logic goes like this:

When Achilles reaches the tortoise’s “initial starting point (Point A)”, the tortoise has moved a little forward and is at “Point B”.
When Achilles reaches “Point B”, the tortoise has moved a little further forward and is at “Point C”.
When Achilles reaches “Point C”, the tortoise has moved a little further forward again and is at “Point D”.

This process continues infinitely. Every time Achilles reaches “where the tortoise was,” the tortoise has inevitably moved “a little bit further ahead.” Although the distance keeps shrinking, because these steps must be repeated an infinite number of times, Achilles can never overtake the tortoise, or so the argument goes.

In the real world, it is obvious that a fast person will overtake a slow person. However, explaining exactly where the flaw lay in this verbal logical trick was extremely difficult for the people of that time.

2. Where is the Error? The Illusion of “Time” and “Infinity”

The cleverness of Zeno’s logic lies in subtly replacing “an infinite number of steps (division of space)” with “infinite time”.

It is true that there are an infinite number of “steps” before Achilles reaches the place where the tortoise was. However, just because “there are an infinite number of steps,” it does not necessarily mean that “the total time required for them is infinite (eternal)”.

Later mathematicians created a powerful weapon called the “sum of an infinite series” to solve this paradox.

3. Mathematical Clarification: The Sum of Infinite Series and “Limits”

Let’s apply concrete numbers to this problem and calculate it mathematically.

Let Achilles’ running speed be $10\text{m}$ per second.
Let the tortoise’s walking speed be $1\text{m}$ per second. ($\frac{1}{10}$ of Achilles’ speed)
As a handicap for the tortoise, assume the tortoise starts $10\text{m}$ ahead of Achilles.

Calculating the Time for Each Step

Step 1: The time it takes for Achilles to reach the tortoise’s initial position ($10\text{m}$ ahead) is $\frac{10\text{m}}{10\text{m/s}} =$ $1\text{ second}$. During this 1 second, the tortoise has moved forward $1\text{m}$. (The current gap between Achilles and the tortoise is $1\text{m}$)

Step 2: The time it takes for Achilles to reach the tortoise’s next position ($1\text{m}$ ahead) is $\frac{1\text{m}}{10\text{m/s}} =$ $0.1\text{ seconds}$. During this 0.1 seconds, the tortoise has moved forward $0.1\text{m}$. (The gap is $0.1\text{m}$)

Step 3: The time it takes for Achilles to reach the tortoise’s next position ($0.1\text{m}$ ahead) is $\frac{0.1\text{m}}{10\text{m/s}} =$ $0.01\text{ seconds}$. During this 0.01 seconds, the tortoise has moved forward $0.01\text{m}$. (The gap is $0.01\text{m}$)

In this way, the “time” it takes for Achilles to reach the tortoise’s previous position forms the following infinite sequence:

$$ 1\text{ second},\ 0.1\text{ seconds},\ 0.01\text{ seconds},\ 0.001\text{ seconds},\ \dots $$

Zeno said, “Since these steps continue infinitely, Achilles can never catch up.” However, what happens if we add up all the time taken for each of these steps (finding the sum of the infinite series)?

$$ \text{Total Time } T = 1 + 0.1 + 0.01 + 0.001 + \dots $$

This is an infinite geometric series with a first term of $a = 1$ and a common ratio of $r = 0.1$. When the absolute value of the common ratio $r$ is less than 1 ($|r| < 1$), the infinite geometric series converges to a certain “finite value.” The formula for its sum is as follows:

$$ S = \frac{a}{1 - r} $$

Calculating this by applying it to the formula gives:

$$ T = \frac{1}{1 - 0.1} = \frac{1}{0.9} = \frac{10}{9} = 1.1111\dots \text{ seconds} $$

In other words, even if there are an infinite number of steps, the sum of the time required for them does not become “infinite,” but converges exactly to $\frac{10}{9}$ seconds (about 1.11 seconds). Achilles will splendidly catch up to and overtake the tortoise approximately 1.11 seconds after the start.

pie title "Time until Achilles catches up (Total approx. 1.11 seconds)" "Step 1 (1 second)" : 90 "Step 2 (0.1 seconds)" : 9 "Infinite sum from Step 3 onwards (0.011... seconds)" : 1

4. Why Were We Deceived?

The essence of this paradox lies in pointing out the flaw in human naive intuition that “if you add up an infinite number of things, the answer must also be infinite.”

$$ 1 + 1 + 1 + 1 + \dots = \infty $$

As shown here, adding the same number infinitely naturally results in infinity.

$$ \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \dots = \infty $$

The famous “harmonic series” also adds numbers that get progressively smaller, but ultimately diverges to infinity.

However, when the numbers being added become smaller sufficiently quickly (like in a geometric series, for example), even if you add an infinite number of them, they neatly fit within a certain “finite boundary.”

$$ \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \frac{1}{16} + \dots = 1 $$

It is the same as how eating half a cake, then half of the remainder, then half of that remainder… infinitely repeating this process will still never exceed “the original 1 whole cake.” Zeno intentionally divided time into minute fragments and, by only speaking within those divided timeframes (1 second, 0.1 seconds, 0.01 seconds…), created the illusion that Achilles could “never catch up.”

5. Easy to Solve in an Instant Using Relative Velocity

By the way, without falling into Zeno’s trap (the infinite division of space and time), it is also easy to solve this problem using middle school mathematics. You just need to use “relative velocity.”

Achilles’ speed: $10\text{m/s}$
Tortoise’s speed: $1\text{m/s}$
The relative speed of the tortoise from Achilles’ perspective (the speed at which Achilles approaches the tortoise): $10 - 1 = 9\text{m/s}$

Achilles’ initial delay relative to the tortoise is $10\text{m}$. The time it takes to close the distance of $10\text{m}$ at a speed of $9\text{m/s}$ is:

$$ \text{Time} = \frac{\text{Distance}}{\text{Speed}} = \frac{10}{9}\text{ seconds} $$

This perfectly matches the answer we previously derived using calculus (the limit of an infinite series).

6. Conclusion: Paradoxes Developed Mathematics

From our modern perspective, Zeno’s “Achilles and the Tortoise” might just seem like wordplay or sophistry. However, for the philosophers of ancient Greece, who lacked concepts like “infinity” and “limits” at the time, refuting this using logic alone was a formidable task.

The deep questions posed by this paradox—“What is continuity?” and “What does it mean to be infinitely divisible?"—became an important driving force leading to the birth of “calculus” by Newton and Leibniz later on, and further to modern mathematical foundations.

Great paradoxes do not merely deceive people; they also act as keys that open doors to new mathematics.

The Two Envelopes Paradox: The Collapse of Logic and Decision-Making Traps Caused by Infinite Expected Values

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

1. The Ultimate Choice: To Switch or Not to Switch?

You are standing on the final stage of a game show. On the table in front of you are two identical-looking envelopes (A and B). The host says to you:

“One envelope contains twice as much money as the other. Please choose one.”

After some hesitation, you choose Envelope A. Just as you are about to look inside, the host whispers the devil’s temptation:

“You can exchange your Envelope A with the remaining Envelope B right now if you want. Would you like to switch?”

Now, should you switch your envelope?

2. The “Infinite Loop” Derived from Expected Value Calculations

Let’s exercise some mathematical thinking here. Suppose the amount in your Envelope A is $X$ yen. According to the rules, the amount in Envelope B is either “half of $X$ yen ($\frac{X}{2}$)” or “twice $X$ yen ($2X$)”. The probability for each is $\frac{1}{2}$ (50%).

Now, let’s calculate the expected value (the estimated average amount) if you switch envelopes.

$$ E = \frac{1}{2} \times \left(\frac{X}{2}\right) + \frac{1}{2} \times (2X) $$ $$ E = \frac{X}{4} + X = \frac{5}{4}X = 1.25X $$

A surprising result emerges. By simply switching envelopes, the expected value jumps to $1.25$ times (a 25% increase) the original $X$ yen. The conclusion becomes, “If you think mathematically, it’s definitely better to switch!”

However, a collapse of logic occurs here. Suppose you switched to Envelope B. What happens if the host asks again right after, “Do you want to switch back to A after all?” The exact same calculation formula applies, and this time it means “Switching from B to A will increase the expected value by 1.25 times.”

In other words, just by continuously switching “from A to B” and “from B to A”, the theoretical expected value will keep increasing infinitely. This clearly contradicts reality (the contents of the envelopes are fixed from the start and do not increase just because you switch them).

graph TD Start["You choose Envelope A (contains X yen)"] --> Think["Calculate if it's profitable to switch"] Think --> Case1["Envelope B has half (X/2 yen) : 50% probability"] Think --> Case2["Envelope B has double (2X yen) : 50% probability"] Case1 --> Calc["Expected Value = (X/4) + X = 1.25X"] Case2 --> Calc Calc --> SwitchToB["Switch to Envelope B! (contains Y yen)"] SwitchToB --> ThinkAgain["Calculate again"] ThinkAgain --> Case3["Envelope A has half (Y/2 yen) : 50% probability"] ThinkAgain --> Case4["Envelope A has double (2Y yen) : 50% probability"] Case3 --> Calc2["Expected Value = 1.25Y"] Case4 --> Calc2 Calc2 --> SwitchToA["Switch back to Envelope A!"] SwitchToA --> Start style Calc fill:#ff9999,stroke:#333,stroke-width:2px style Calc2 fill:#ff9999,stroke:#333,stroke-width:2px style SwitchToA fill:#ff4444,color:#fff,stroke:#333,stroke-width:4px

Why did a seemingly perfect expected value calculation produce such a strange paradox?

3. Demystifying the Mathematical Trick: The Swap of Variables

The trap of this paradox lies in “how the random variable $X$ is used”.

In the previous calculation formula, we treated the amount $X$ in Envelope A as a fixed constant, and assumed Envelope B is either “$\frac{X}{2}$ or $2X$”. However, what is actually fixed is the “total amount of money in the two envelopes”, or the “smaller amount”.

Let $S$ be the amount in the envelope with less money. Then, the envelope with more money contains $2S$. There are only two possible scenarios for the entire game (the probability of each is $\frac{1}{2}$).

Pattern 1: Envelope A you chose has the smaller amount ($S$), and Envelope B has the larger amount ($2S$)
Pattern 2: Envelope A you chose has the larger amount ($2S$), and Envelope B has the smaller amount ($S$)

Now, let’s correctly calculate the expected values for “not switching” and “switching” the envelopes.

Expected value when not switching $E_{stay}$:

$$ E_{stay} = \frac{1}{2} \times S + \frac{1}{2} \times 2S = \frac{3}{2}S = 1.5S $$

Expected value when switching $E_{switch}$: You get $2S$ in Pattern 1, and $S$ in Pattern 2.

$$ E_{switch} = \frac{1}{2} \times 2S + \frac{1}{2} \times S = \frac{3}{2}S = 1.5S $$ $$ E_{stay} = E_{switch} $$

The expected values match perfectly! In the first incorrect calculation, we treated the $X$ in Pattern 1 (which is actually $S$) and the $X$ in Pattern 2 (which is actually $2S$) as different values using the same variable $X$, which created the illusion that “switching increases the expected value.”

pie title "The Truth of Expected Values (assuming the smaller amount is S)" "Expected value of not switching (1.5S)" : 50 "Expected value of switching (1.5S)" : 50

4. What If You Open the Envelope?

The paradox seems to be resolved. However, a deeper problem awaits.

What if you looked inside your Envelope A before exchanging envelopes? When you open Envelope A, you find “10,000 yen” inside.

At this moment, $X = 10000$ becomes a fixed value. Envelope B contains either “5,000 yen” or “20,000 yen”. What happens if we apply the very first calculation formula here?

$$ E_{switch} = \frac{1}{2} \times 5000 + \frac{1}{2} \times 20000 = 2500 + 10000 = 12500 $$

The expected value is 12,500 yen. It is certainly higher than the current 10,000 yen. Moreover, since $X$ is now a “specific constant” of 10,000 yen, the previous counterargument of the “swap of variables” no longer applies. In this case, is it absolutely better to switch?

The Disproof by Bayesian Inference: The Missing “Prior Distribution”

In response to this, mathematicians introduced the concept of the “prior distribution of amounts (prior probability)”. The question is whether we can truly say that 5,000 yen and 20,000 yen are each inside with a $\frac{1}{2}$ probability.

For example, suppose the maximum budget for the show is 100 million yen. If you open Envelope A and find “60 million yen”, the probability that Envelope B contains “120 million yen” is zero (because it’s over budget). In other words, as the amount in Envelope A gets larger, the probability that Envelope B is “double” must decrease, and the probability that it is “half” must increase.

When calculating the expected value using Bayes’ theorem assuming an arbitrary prior distribution $P(x)$, it has been mathematically proven that under any realistic probability distribution (where the sum is 1), there is no magical distribution that makes it “better to switch” for all amounts of $X$.

5. The Infinite Trap: Connection to the St. Petersburg Paradox

There is only one case where it is “better to switch for all $X$”. That is only if we assume the show’s budget is infinite and all amounts (1 yen, 2 yen, 4 yen, 8 yen… up to infinity) appear uniformly—an “improper prior distribution” (a distribution whose sum is infinity).

However, in the real world, no television station has infinite assets. The bug caused by this “infinite expected value” shares deep roots with the St. Petersburg paradox (the problem of how much a person would be willing to pay for a gamble with an infinite expected value).

6. Conclusion: The Terrors of Probability and Expected Value

Even though the “Two Envelopes Paradox” consists only of simple multiplication and addition, it teaches us the following lessons:

Errors caused by ambiguity in definitions: If you do not clarify what a variable refers to (whether $X$ always refers to the same amount), logic can easily collapse.
The illusion of “no information = 50% probability”: The assumption that “because we don’t know, it must be fifty-fifty” (the principle of insufficient reason) can sometimes lead to fatal miscalculations.
The difficulty of handling infinity: Introducing the concept of “infinity,” which cannot be applied to the real world, into calculation formulas produces results that defy common sense.

The next time in life you think, “The grass is greener on the other side, so it’s better to switch,” remember this paradox. In your calculation formula, the variables might just be getting swapped.