Galton's Paradox: What is the probability that three coins will show the same side?

[Wise Young]

Suppose you have three pennies and you flip each of the coins. What is the probability that all three coins will show the same face? The answer to this question is Galton's paradox and the answer is contained in Source.

[Le Todd]

That makes sense.

[rfbdorf]

Problem is, that source gives the incorrect answer! That's why it's a paradox - the 'obvious' answer is incorrect.

I had not seen this one before, but here's how I approached it: The listing of all possible outcomes is
HHH, HHT, HTH, HTT, THH, THT, TTH, TTT
There are 8 possible outcomes; 2 of those are all heads or all tails. So the probability of all the same side is 2/8 = 1/4. That's the correct answer.

As nicely explained by Francis Galton himself in his very readable paper in Nature (1894), the fallacy lies in confusing a particular coin with any coin. The argument goes:
1) At least 2 of the coins must turn up alike.
2) It is an even chance whether A third coin is heads or tails
3) Therefore, it is an even chance whether THE third coin is heads or tails.

Wrong! 'A third coin'; is not the same as 'the third coin'!

Looking at the 8 possible outcomes, you can see that there are 4 ways for any 2 coins of the 3 to be both heads: HHH, HHT, HTH, and THH. But the third coin is, respectively, H,T,T,T - that is, there is only a 1/4 chance - not 1/2 - that the third coin is the same as the 2 coins that agree! That's because the third coin is actually not one particular coin, but may be any of the three - hence the confusion I pointed out between 'A' and 'THE.'

This is really a neat puzzle. Galton was quite a guy. He was also half cousin to Charles Darwin.
- Richard

[Foolish Old]

rfbdorf has given the correct answer (2/8), and a very cogent explanation. Each of the coin flips is an independent event.

[Le Todd]

Richard,

Does the probability change based on how you approach the puzzle? Isn't it considered a paradox because if you look at the puzzle simply based on the basic outcome, you think it's 1/2, but upon further examination you consider each coin its own entity rather than the puzzle as a whole?

[rfbdorf]

Exactly. It's called a paradox because the flaw in the argument leading to the 1/2 result is not at all easy to discover. The argument that Galton shows (and deconstructs) is very cleverly constructed to be obfuscatory. I do recommend reading his short paper.
- Richard

[DougM]

The flaw as I see it is that you have to assume that the first two coins land on matching sides. That's not necessarily true and the explanation they provide assumes it is.