The only one I ever actually learned here was Simpson's paradox.

Vote
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"You will never reach point B from point A as you must always get half-way there, and half of the half, and half of that half, and so on."
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### Hilbert's paradox of the Grand Hotel

If a hotel with infinitely many rooms is full, it can still take in more guests.
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### Monty Hall problem

An unintuitive consequence of conditional probability.
1 vote
1 comment
4

For almost everyone, their friends have more friends than they do.
1 vote
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Cut a ball into a finite number of pieces, re-assemble the pieces to get two balls, both of equal size to the first.
1 vote
1 comment
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A test that is accurate the vast majority of the time could show you have a disease, but the probability that you actually have it could still be tiny.
1 vote
1 comment
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Why one will wait longer for a bus than one should.
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A two-child family has at least one boy. What is the probability that it has a girl?
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### Will Rogers phenomenon

The mathematical concept of an average, whether defined as the mean or median, leads to apparently paradoxical results—for example, it is possible that moving an entry from an encyclopedia to a dictionary would increase the average entry length on b... oth books   more
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There is no greatest cardinal number.
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You are given two indistinguishable envelopes, each of which contains a positive sum of money. One envelope contains twice as much as the other. You may pick one envelope and keep whatever amount it contains. You pick one envelope at random but befo... re you open it you are given the chance to take the other envelope instead   more
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A wager between two people seems to favour them both. Very similar in essence to the Two-envelope paradox.
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Bertrand's box paradox is a classic paradox of elementary probability theory. It was first posed by Joseph Bertrand in his Calcul des probabilités, published in 1889.There are three boxes:a box containing two gold coins,a box containing two silver c... oins,a box containing one gold coin and one silver coin.After choosing a box at random and withdrawing one coin at random, if that happens to be a gold coin, it may seem that the probability that the remaining coin is gold is ¹⁄₂; in fact, the probability is actually ²⁄₃. Two problems that are very similar are the Monty Hall problem and the Three Prisoners problem.These simple but counterintuitive puzzles are used as a standard example in teaching probability theory. Their solution illustrates some basic principles, including the Kolmogorov axioms   more
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A trend that appears in different groups of data disappears when these groups are combined, and the reverse trend appears for the aggregate data.
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The Kelly criterion is an often optimal strategy for maximizing profit in the long run. Proebsting's paradox apparently shows that the Kelly criterion can lead to ruin.
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### All horses are the same color

The horse paradox is a falsidical paradox that arises from flawed demonstrations, which purport to use mathematical induction, of the statement All horses are the same color. There is no actual contradiction, as these arguments have a crucial flaw t... hat makes them incorrect. This example was used by Joel E. Cohen as an example of the subtle errors that can occur in attempts to prove statements by induction   more
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The number of points of intersection of two higher-order curves can be greater than the number of arbitrary points needed to define one such curve.
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Elevators can seem to be mostly going in one direction, as if they were being manufactured in the middle of the building and being disassembled on the roof and basement.
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The first number that can be considered "dull" rather than "interesting" becomes interesting because of that fact.
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### Nontransitive dice

You can have three dice, called A, B, and C, such that A is likely to win in a roll against B, B is likely to win in a roll against C, and C is likely to win in a roll against A.
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If you let potatoes consisting of 99% water dry so that they are 98% water, they lose 50% of their weight.
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Does the set of all those sets that do not contain themselves contain itself?
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Effect size may not be indicative of practical meaning.
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Predictive models with a given level of accuracy may have greater predictive power than models with higher accuracy.
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### Benford's law

Numbers starting with lower digits appear disproportionately often in seemingly random data sets.
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Berkson's paradox also known as Berkson's bias or Berkson's fallacy is a result in conditional probability and statistics which is counterintuitive for some people, and hence a veridical paradox. It is a complicating factor arising in statistical te... sts of proportions. Specifically, it arises when there is an ascertainment bias inherent in a study design. It is often described in the fields of medical statistics or biostatistics, as in the original description of the problem by Joseph Berkson   more
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Describes a problem in model selection where predictor variables with no explanatory power can appear artificially important.
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A sphere can, topologically, be turned inside out.
29

### Missing square puzzle

Two similar-looking figures appear to have different areas while built from the same pieces.
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There exists a countable subset C of the sphere S such that S\C is equidecomposable with two copies of itself.
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After alternatively adding and removing balls to a vase infinitely often, how many balls remain?
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### Thomson's lamp

After flicking a lamp on and off infinitely often, is it on or off?
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A coin rotating along the edge of an identical coin will make a full revolution after traversing only half of the stationary coin's circumference.
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Though most numbers are not squares, there are no more numbers than squares.
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Shows that the exact meaning of statements involving conditionals and probabilities is more complicated than may be obvious on casual examination.
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Tiny errors in the null hypothesis are magnified when large data sets are analyzed, leading to false but highly statistically significant results.
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Different common-sense definitions of randomness give quite different results.
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Countably infinite models of set theory contain uncountably infinite sets.
39

### Three cards problem

When pulling a random card, how do you determine the color of the underside?
40

### Three Prisoners problem

The Three Prisoners problem appeared in Martin Gardner's "Mathematical Games" column in Scientific American in 1959. It is mathematically equivalent to the Monty Hall problem with car and goat replaced with freedom and execution respectively, and al... so equivalent to, and presumably based on, Bertrand's box paradox   more
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Apparently, a man can be "forced to stay where he is by the mere unfulfilled intentions of the gods".
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What is the chance that two people in a room have the same birthday?
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Low birth weight and mothers who smoke contribute to a higher mortality rate. Babies of smokers have lower average birth weight, but low birth weight babies born to smokers have a lower mortality rate than other low birth weight babies. This is a sp... ecial case of Simpson's paradox   more
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### Sleeping Beauty problem

The Sleeping Beauty problem is a puzzle in probability theory and formal epistemology in which an ideally rational epistemic agent is to be woken once or twice according to the toss of a coin, and asked her degree of belief for the coin having come ... up heads.The problem was originally formulated in unpublished work by Arnold Zuboff, followed by a paper by Adam Elga but is based on earlier problems of imperfect recall and the older "paradox of the absentminded driver". The name Sleeping Beauty for the problem was first used in extensive discussion in the Usenet newsgroup rec.Puzzles in 1999   more
45

### I don't understand.

We use numbers every day, but taking a step back, what are they, really — and why do they do such a damn good job of helping us explain the universe (such as Newtonian laws)? Mathematical structures can consist of numbers, sets, groups, and points —...  but are they real objects, or do they simply describe relationships that necessarily exist in all structures? Plato argued that numbers were real (it doesn't matter that you can't "see" them), but formalists insisted that they were merely formal systems (well-defined constructions of abstract thought based on math). This is essentially an ontological problem, where we're left baffled about the true nature of the universe and which aspects of it are human constructs and which are truly tangible   more
46

### Gabriel's Horn

A simple object with finite volume but infinite surface area. Also, the Mandelbrot set and various other fractals are covered by a finite area, but have an infinite perimeter (in fact, there are no two distinct points on the boundary of the Mandelbr... ot set that can be reached from one another by moving a finite distance along that boundary, which also implies that in a sense you go no further if you walk "the wrong way" around the set to reach a nearby point). This can be represented by a Klein bottle   more
47

### Nikodym set

A set contained in and with the same Lebesgue measure as the unit square, yet for every one of its points there is a straight line intersecting the Nikodym set only in that point.
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A set that can be partitioned into two sets, each of which is equivalent to the original.
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The perimeter of a landmass is in general ill-defined.
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If the ordinal numbers formed a set, it would be an ordinal number that is smaller than itself.
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