Solve coin probability and Bayesian inference problems
Company: Five Rings
Role: Data Scientist
Category: Statistics & Math
Difficulty: hard
Interview Round: Take-home Project
Answer the following coin-related statistics questions:
1) You repeatedly toss a fair coin. What is the probability that the pattern HHH appears before the pattern THH?
2) You toss a fair coin 15 times. What is the expected length of the longest consecutive run of heads?
3) Let θ be the probability of heads for a coin with prior density p(θ) ∝ θ^3 for θ ∈ [0, 1]. After observing 4 tosses, all of which are heads, what is the posterior distribution of θ?
Quick Answer: This question evaluates probabilistic reasoning, stochastic-process intuition (pattern waiting times and runs), expected-value calculations, and Bayesian parameter estimation competency.