Coin Rotation Paradox

The coin rotation paradox is a famous math problem with an unintuitive solution:

If you roll a coin around the edge of another coin of the same size, from an external perspective, how many rotations does the coin make by the time it returns to its original position?

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Boy or Girl Paradox

The “boy or girl paradox” is a well-known brain teaser by famous puzzle-maker Martin Gardner. It’s popularly considered a “paradox” because (1) it has a highly unintuitive solution, and (2) its ambiguous wording meant either of two solutions could be valid solutions.

This is a rewording of that brain teaser to eliminate some ambiguity from that original question:

Out of all families with exactly two children, we randomly pick one family that has at least one boy. What is the probability that both children in this family are boys?

Assume only for the purposes of this puzzle that a child can only be a boy or a girl, and that either possibility is equally likely.

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Two Envelopes Paradox

The two envelopes paradox is a famous brain teaser of sorts, and not a true paradox. The problem is generally posed like this:

You are given a choice between two identical envelopes. One envelope contains some amount of money, and the other contains twice that amount of money. There is no way to distinguish between the two. However, when you choose one of the envelopes, before opening it, you are given the option of switching to the other envelope. Should you switch?

Why is this sometimes called a paradox? Well, if you choose to switch, you have a 50% chance of doubling your money, and a 50% chance of halving your money. If the amount of money in the envelope you initially chose is M, this reasoning suggests the expected amount in the other envelope is (2M + 0.5M) / 2 = 1.25M. This is more than M, so you should always switch.

But that would suggest once you’ve switched, you’re in the same position you were before you switched, so you should switch again. What is the problem with this reasoning?

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The Monty Hall Problem Explained

The Monty Hall Problem

You are on a game show in which there are three identical doors, one with a car behind it and two with goats behind them. You must pick one door, and you win if that door has the car behind it.

After you pick a door, the host of the game show always opens a door you didn’t choose that has a goat behind it. This leaves the door you chose and one other remaining door, and you are given the option to switch your choice to the other remaining door.

Should you switch or should you stick to your original choice? What chance of winning would that give you?

The History

The Monty Hall Problem is a classic probability puzzle, named for its similarity to the game show “Let’s Make a Deal”, which was hosted by Monty Hall. The problem was made famous when Marilyn vos Savant answered it correctly in her column in a popular magazine, and thousands of readers wrote letters to the magazine arguing her solution was wrong!

The solution can be counter-intuitive, so give it some thought and then scroll down to see the Monty Hall Problem explained.

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Berkson’s Paradox

Berkson’s Paradox is a counterintuitive or unexpected trend observed in a sample due to a particular type of selection bias. This bias arises when the sample is selected based on the combination of two characteristics.

Also known as Berkson’s bias or collider bias, Berkson’s Paradox pertains to situations where a group is selected based on the combination of two characteristics and results in some false observation of correlation between the two characteristics – the correlation might be observed in the sample only because those without those two characteristics were not selected to be in the group in the first place.

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Base Rate Fallacy

What is the Base Rate Fallacy?

In simple terms, it’s a common error we make in assessing likelihoods due to (a) over-emphasizing the rate of something within a group and (b) under-emphasizing how common that group is in the first place (i.e., the base rate).

For example, let’s say you see a chess set in a building with 1 avid chess player and 1000 other people. You might assume it belongs to the chess player, even though it’s more likely to belong to one of the others because there are so many of them – if only 1% of regular people own chess sets, there would likely be ~10 of them in a group of 1000, outnumbering the 1 chess player.

Sometimes also referred to as Base Rate Bias or Base Rate Neglect, this is a cognitive bias arising from the tendency to place too much emphasis on event-specific information, at the expense of relevant base rate information. Often this results in a sense of probabilities or rates that are very far from reality!

To understand what this means, let’s look at a few more examples:

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