There are two coins in a bag: - a coin has heads on both sides;
- the other is a normal coin, with heads and tails. You draw one at random, toss it, and it comes up heads. What is the probability that the other side of that same coin is also heads?
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The coin of two heads
It is a small and very clean paradox: a coin, a toss and new information are enough for intuition to begin to distribute the probabilities incorrectly. The good reader here starts again from the observed data.
Hints
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- After seeing heads, not all initial possibilities continue to weigh equally.
- Don't count coins; count results compatible with what you have seen.
Solution
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Answer: $\tfrac{2}{3}$. When conditioning on “heads,” the equiprobable cases are these three observations: 1. double-sided coin, side 1,
- double-sided coin, side 2,
- normal coin, face side. In 2 out of 3 cases you are in the double-sided coin. Reusable idea: in conditional probability you have to reconstruct the sample space after the evidence.
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