Bayes' Theorem with Coins

Update beliefs about coins based on evidence

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Biased Coin DetectionBiased?Which Coin?Which?Fairness TestTest

Sequential Fairness Testing

The Scenario

You want to test if a coin is fair. You flip it multiple times and update your belief after each flip. This demonstrates how Bayesian updating works in real-time.

Initial Assumptions

Fair Coin Hypothesis

(prior)

Biased Coin Hypothesis

(prior)

Sequence of Flips: H, H, T, H

Flip 1: Heads

Update belief after seeing heads:

P(fair)

40%

P(biased)

60%

Flip 2: Heads

Update using new prior (40% fair):

P(fair)

30.8%

P(biased)

69.2%

Flip 3: Tails

Tails favors the fair coin:

P(fair)

47.1%

P(biased)

52.9%

Flip 4: Heads

Back to heads:

P(fair)

37.3%

P(biased)

62.7%

Belief Evolution Summary

Flip	Result	P(Fair)	P(Biased)
Before	-	50.0%	50.0%
1	H	40.0%	60.0%
2	H	30.8%	69.2%
3	T	47.1%	52.9%
4	H	37.3%	62.7%

Key Insights

1.Each flip updates our belief based on the previous posterior
2.Heads pushes belief toward biased; Tails pushes toward fair
3.With 3 heads and 1 tail (75% heads), belief slightly favors biased
4.More flips provide more certainty either way