Bayes' Theorem with Coins

Update beliefs about coins based on evidence

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Biased Coin DetectionBiased?Which Coin?Which?Fairness TestTest

Which Coin Did I Flip?

Fair Coin

P(H) = 0.5

Biased Coin

P(H) = 0.7

The Scenario

You have two coins: a fair one (50% heads) and a biased one (70% heads). You pick one at random and flip it once, getting heads. What is the probability you picked the fair coin?

Problem Setup

Fair Coin

Biased Coin

Step-by-Step Solution

Calculate P(Heads) - Total Probability

Apply Bayes' Theorem

Answer

Given you flipped heads, there is a 41.7% chance you used the fair coin, and a 58.3% chance you used the biased coin.

P(Fair | Heads)

41.7%

P(Biased | Heads)

58.3%

What if You Got Tails?

If you had flipped tails instead, the calculation would favor the fair coin:

Tails is more likely from the fair coin (50%) than the biased coin (30%), so tails is evidence for the fair coin!