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.    

Galton's Paradox
Francis Galton’s Paradox

 

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:

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