Posterior Probability of a Two-Headed Coin After Three Heads

Coin A is unbiased, coin B lands heads with chance 0.8, and coin C has heads on both sides. A coin is chosen uniformly at random from these three and flipped three times. Given that all three flips are heads, what is the probability that coin C was selected?

Hint — Evaluate evidence coin by coin: For each coin, what is the likelihood of seeing three heads if that coin were the one drawn? Combine those likelihoods with the probability of selecting each coin in the first place.

Constraints and Clarifications

• The flips are independent conditional on the chosen coin.
• Each coin has prior probability $\tfrac{1}{3}$.
• Give an exact fraction and a decimal approximation.

What a Strong Answer Covers

• Clearly stated priors and the probability of three heads under each coin.
• Correct use of Bayes' rule, including the total probability of the observed data.
• An exact result together with a sanity check against coin C's prior.

Follow-up Questions

1. Given those same three heads, what is the probability that a fourth flip of the same coin lands heads?
2. How many consecutive heads are required for the posterior probability of coin C to exceed 0.95?
3. How would the answer change if exactly one of the three flips were tails? Overview: A Bayesian probability problem with three coins: one fair, one that lands heads with probability 0.8, and one double-headed. After a uniformly chosen coin shows heads on all three tosses, find the probability that the two-headed coin was picked, testing Bayes' rule, likelihoods, and normalization.