Posterior Probability of a Two-Headed Coin After Three Heads

Coin A is fair, coin B lands heads with probability 0.8, and coin C has heads on both sides. One of the three coins is selected uniformly at random and tossed 3 times. Given that all 3 tosses land heads, what is the probability that coin C was selected? ```hint Weigh the evidence per coin For each coin, how likely is the observed outcome if that coin was the one selected? Combine those quantities with how likely each coin was to be selected in the first place. ``` ### Constraints and Clarifications - The tosses are independent given the selected coin. - Each coin is selected with probability $\tfrac{1}{3}$. - Give an exact fraction and a decimal approximation. ### What a Strong Answer Covers - Clearly defined prior probabilities and the likelihood of three heads under each coin. - A correct application of Bayes' rule, including the total probability of the observation. - An exact result together with a sanity check against the prior probability of coin C. ### Follow-up Questions 1. Given the same three heads, what is the probability that a fourth toss of the same coin lands heads? 2. How many consecutive heads would be needed before the probability that coin C was selected exceeds 0.95? 3. How would the answer change if exactly one of the three tosses had landed tails?