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Solve coin probability and Bayesian inference problems
Coin-Flip Statistics: Pattern Racing, Runs, and Bayesian Updating
This problem presents three independent questions related to fair coin tossing and Bayesian inference. Assume all coin flips are independent events.
Tasks
- Pattern Racing Probability: Repeatedly toss a fair coin until either the pattern
HHHor the patternTHHappears. Calculate the probability thatHHHappears beforeTHH. - Expected Longest Run: Toss a fair coin 15 times. Determine the expected length of the longest consecutive run of heads.
- Bayesian Posterior Distribution: Consider a coin with an unknown probability of heads,
θ. The prior density forθis given byp(θ) ∝ θ^3forθ ∈ [0, 1]. After observing 4 tosses, all of which are heads, derive the posterior distribution ofθ.
AI interview prep
Core dilemma
The central challenge is to accurately apply diverse probabilistic and statistical methodologies, ranging from combinatorial probability and Markov chains to Bayesian inference, demonstrating both theoretical understanding and practical problem-solving skills.
Mid-level signals3
Senior signals3
Red flags3
2 tradeoffs and 3 stress tests locked
Interview prep
Unlock the interviewer X-Ray rubric, tradeoff matrix, stress tests, clarifying questions, and key concepts.
Solve coin probability and Bayesian inference problems
Coin-Flip Statistics: Pattern Racing, Runs, and Bayesian Updating
This problem presents three independent questions related to fair coin tossing and Bayesian inference. Assume all coin flips are independent events.
Tasks
- Pattern Racing Probability: Repeatedly toss a fair coin until either the pattern
HHHor the patternTHHappears. Calculate the probability thatHHHappears beforeTHH. - Expected Longest Run: Toss a fair coin 15 times. Determine the expected length of the longest consecutive run of heads.
- Bayesian Posterior Distribution: Consider a coin with an unknown probability of heads,
θ. The prior density forθis given byp(θ) ∝ θ^3forθ ∈ [0, 1]. After observing 4 tosses, all of which are heads, derive the posterior distribution ofθ.