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Posterior Probability of a Two-Headed Coin After Three Heads

DRWQuantitative Researcher · Onsite
HardQuantitative ResearcherOnsiteStatistics & Math

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This question tests the application of Bayes' theorem, conditional likelihoods, and posterior probability updates given sequential binary observation data.

Problem Statement

You are presented with a pouch containing three coins with identical physical appearances but different weightings:

  • Coin A (Fair Coin): Lands on Heads with probability P(H | A) = 0.5.
  • Coin B (Biased Coin): Lands on Heads with probability P(H | B) = 0.8.
  • Coin C (Two-Headed Coin): Lands on Heads with probability P(H | C) = 1.0.

You draw one coin uniformly at random from the pouch without inspecting it. You then flip this chosen coin 3 times independently, and all 3 tosses land on Heads.

Tasks

  1. Calculate the exact posterior probability that the chosen coin is the two-headed coin (Coin C), expressed as both an irreducible fraction and a decimal rounded to four decimal places.
  2. Calculate the posterior probability distribution over all three coins: P(A | 3H), P(B | 3H), and P(C | 3H).
  3. Using the updated posterior distribution, determine the probability that a fourth toss of this same coin will also result in Heads.

Assumptions

  • The initial draw is strictly uniform: P(A) = P(B) = P(C) = 1/3.
  • Coin tosses are conditionally independent given the identity of the coin.

Hints

Hint 1

Write down the prior probability for each coin and compute the likelihood of observing three consecutive heads for each individual coin.

Hint 2

Use the law of total probability to sum the weighted likelihoods across all three hypotheses for the denominator.

Hint 3

For the fourth toss, compute the expected probability of Heads by weighting each coin's P(H) by its updated posterior probability.

Follow-up questions

  • How many consecutive heads `n` would you need to observe to be at least 99% confident that you picked the two-headed coin?
  • Suppose the prior was not uniform, but instead biased towards the fair coin such that `P(A) = 0.8`, `P(B) = 0.15`, and `P(C) = 0.05`. How does the posterior probability of Coin C change after 3 heads?
  • If the third toss had landed on Tails instead of Heads, how would you instantly update the posterior probabilities without full recalculation?