Galton’s Paradox supposes you have three fair coins. Necessarily, two sides will match (ie: two will be heads, or two will be tails). It’s an even probability that the third coin will be a head or tail. Therefore, the chance that all three will match is 1/2. Is our solution correct? Quite

Galton’s Paradox supposes you have three fair coins. Necessarily, two sides will match (ie: two will be heads, or two will be tails). It’s an even probability that the third coin will be a head or tail. **Therefore, the chance that all three will match is 1/2. **

## Is our solution correct?

Quite obviously not, or it wouldn’t be much of a paradox…

Francis Galton, in his book 1894 paper, noted that the fallacy lies in confusing a *particular* coin with *any* coin.

Here’s the fallacy in action:

- At least 2 of the coins must turn up alike.
- It is an 1/2 chance whether a third coin is heads or tails.
- Therefore, it is a 1/2 chance whether the 3rd coin is heads or tails.

Wrong! “**A third coin**” is not the same as “**the third coin**“.

Let’s look at all the possible outcomes of flipping three coins to get the real answer. There are 2^3 possible outcomes:

H H H

H H T

H T H

H T T

T H H

T H T

T T H

T T T

The *original* claim said that the chance that all three outcomes would match was 1/2 (or 50%). But of the 8 possible outcomes, *one* of them is all heads (H H H) **and** *one* of them is all tails (T T T). Therefore, **2 of 8**, or **25%**, results in our desired outcome of all three coins getting the same side.

Tricky.