Compute posterior and predictive coin probabilities
Problem Description
You are presented with a scenario involving two distinct coins:
- Fair Coin (F): Probability of Heads P(H) = 0.5
- Biased Coin (B): Probability of Heads P(H) = 0.7
One of these coins is chosen uniformly at random, meaning P(F) = P(B) = 0.5. The chosen coin will be used for all subsequent flips.
You observe the first two flips of the chosen coin, and both are Heads (H, H). The outcome of the third flip is not yet known.
Tasks
(a) Compute the posterior probability that the chosen coin is the Biased coin, given that the first two flips were Heads: P(B | first two flips are Heads).
(b) Compute the predictive probability that the next (third) flip will be Heads, given that the first two flips were Heads: P(next flip is Heads | first two flips are Heads).
(c) Continuing with the same chosen coin, let T be the total number of flips required until the first Tail occurs (this count includes the flip that results in a Tail). Compute the expected value of T, given that the first two flips were Heads: E[T | first two flips are Heads]. You must show your complete derivation for this calculation.
AI interview prep
Core dilemma
Accurately updating probabilistic beliefs about an underlying state (the chosen coin's bias) based on observed evidence, and subsequently using these updated beliefs to make predictions and calculate expectations for future events.
3 tradeoffs and 3 stress tests locked
Interview prep
Unlock the interviewer X-Ray rubric, tradeoff matrix, stress tests, clarifying questions, and key concepts.
Compute posterior and predictive coin probabilities
Problem Description
You are presented with a scenario involving two distinct coins:
- Fair Coin (F): Probability of Heads P(H) = 0.5
- Biased Coin (B): Probability of Heads P(H) = 0.7
One of these coins is chosen uniformly at random, meaning P(F) = P(B) = 0.5. The chosen coin will be used for all subsequent flips.
You observe the first two flips of the chosen coin, and both are Heads (H, H). The outcome of the third flip is not yet known.
Tasks
(a) Compute the posterior probability that the chosen coin is the Biased coin, given that the first two flips were Heads: P(B | first two flips are Heads).
(b) Compute the predictive probability that the next (third) flip will be Heads, given that the first two flips were Heads: P(next flip is Heads | first two flips are Heads).
(c) Continuing with the same chosen coin, let T be the total number of flips required until the first Tail occurs (this count includes the flip that results in a Tail). Compute the expected value of T, given that the first two flips were Heads: E[T | first two flips are Heads]. You must show your complete derivation for this calculation.