## What is a paradox?

There are multiple definitions of what exactly a paradox is, since the word has been used in different ways. Here is one simplified take on distinguishing types of paradoxes:

#### A. Self-contradicting statement or proposition

This is a logical paradox, something that contradicts itself and therefore cannot be proven true or false. This is sometimes referred to as a “true” paradox or an “antinomy.”

An example is the liar paradox:

This statement is false

If the above statement were true, then its claim would make it false. And if it were false, then its claim would make it true. There is no way to resolve a true logical paradox like this.

These are also statements that *appear *to be contradictory, even though there is a way to resolve it such that there is no logical contradiction.

#### B. Highly unintuitive phenomenon

These paradoxes contain no contradiction at all. However, the situation or claim is so contrary to expectations, that it seems absurd or impossible.

For an example, see the missing dollar paradox (spoiler alert: there isn’t really a missing dollar, but the way the problem is presented leads you into thinking there is).

Sometimes these types of paradoxes are divided into two categories:

- Veridical paradoxes – a statement that appears to be absurd but is actually true
- Falsidical paradoxes – a statement that appears to be absurd, even though there seems to be logic that explains it, but is actually false (usually due to a fallacy in the logic)

Paradoxes on Brain Easer are mostly this type, because they make for interesting puzzles!

## Some interesting paradoxes

Paradoxes can be mind-bending stuff. Most of the paradoxes below require some high-school level knowledge of probability to understand.

#### Two Envelopes Paradox

If there are two identical envelopes, one with twice as much money as the other, and you choose one envelope, should you switch? Calculating the expected value appears to suggest switching is always a good idea…

#### Boy or Girl Paradox

If we pick a family of two children, of which at least one is a boy, what is the probability both are children? Seems simple, but the answer may be unintuitive!

#### The Monty Hall Problem

A classic. Three identical doors, but only one has the prize behind it. After you pick a door, the host reveals one other door has no prize, then offers you the chance to switch to the last door. Should you take the offer?

#### False Positive Paradox

A drunkenness test that has a 5% chance of falsely accusing someone could actually result in false accusations of over 90%, if the prevalence of drunks within the population is low enough!

#### Simpson’s Paradox

I can have higher grades than you in two separate semesters, but somehow have a *lower *average grade across both semesters combined!

#### Berkson’s Paradox

If I select a sample of people based on a combination of two characteristics, you might notice completely misleading and incorrect trends when looking at people in that sample.

#### Coin Rotation Paradox

If you roll a coin around the edge of an identical coin, how many rotations does the coin make when it returns to its starting position? You might think it should be one rotation, but you’d be wrong.