The 1,000-Coin Bayes Puzzle

You pick 1 coin uniformly at random from a bag with 1 double-headed coin and 999 fair coins. After flipping the chosen coin and seeing only heads, what is the probability it is the double-headed coin?

Interactive evidence

Consecutive heads observed: 10

Prior

P(D) = 1/1000

Likelihood

P(H^10 | F) = 1/1024

Posterior

1024 / 2023

Answer

50.62%

After 10 heads in a row, the coin is slightly more likely than not to be the double-headed one.

Bayes' Rule

P(D | Hⁿ) =

[ P(Hⁿ | D) · P(D) ] / [ P(Hⁿ | D) · P(D) + P(Hⁿ | F) · P(F) ]

For n = 10:

P(D | H¹⁰) = [ 1 · (1/1000) ] / [ 1 · (1/1000) + (1/1024) · (999/1000) ]

= 1024 / 2023 ≈ 50.62%

Probability coin is double-headed 50.62%

Probability coin is fair 49.38%

Intuition:
The double-headed coin starts with a tiny prior chance: only 1/1000. But seeing 10 heads in a row is 1024 times more likely under a double-headed coin than under a fair coin. That evidence almost exactly cancels the huge prior disadvantage.