Bayes' Theorem with Coins

Update beliefs about coins based on evidence

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Biased Coin DetectionBiased?Which Coin?Which?Fairness TestTest

Is the Coin Biased?

The Bayesian question:

We observe evidence (8 heads out of 10 flips) and want to know: how should this change our belief about whether the coin is biased?

Why not just conclude "it's biased"?

Getting 8 heads in 10 flips is unlikely but possible with a fair coin. Bayes' theorem helps us weigh the evidence against our prior knowledge (most coins are fair) to reach a reasoned conclusion.

The Scenario

You have a coin that might be biased. You flip it 10 times and get 8 heads. If the coin were biased, it would have P(heads) = 0.8. What is the probability the coin is actually biased?

Prior Assumption

Assume that before flipping, we believe there's a 10% chance any random coin is biased. This is our "prior" — our starting belief before seeing evidence.

Setting Up the Problem

Fair Coin

Biased Coin

Binomial Probability

The probability of getting exactly 8 heads in 10 flips follows the binomial distribution:

Step-by-Step Solution

Calculate P(8 heads | biased)

Calculate P(8 heads | fair)

Calculate P(8 heads) - Total Probability

Apply Bayes' Theorem

Answer

After observing 8 heads in 10 flips, the probability that the coin is biased increased from 10% to approximately 43.3%.

Prior (before flipping)

10%

Posterior (after 8 heads)

43.3%

Key Insight

Even though 8/10 heads seems like strong evidence for a biased coin, we're still more likely to have a fair coin (56.7% vs 43.3%)! This is because:

1.Fair coins are 9x more common (90% vs 10%)
2.8/10 heads can still happen with a fair coin (probability ~4.4%)
3.More flips would provide stronger evidence