Price a coin-doubling game rationally

Consider the following game: you begin with $1. A fair coin is tossed repeatedly. Each time it lands Heads, your bankroll doubles; the first time it lands Tails, the game stops and you collect your current bankroll. For example, H,H,T pays $4. a) Compute the expected payout and say whether it converges; show your derivation. b) If you have to give the most you would pay to play once, assuming risk-averse log utility $U(w)=\ln(w)$ and initial wealth $W$, what price do you choose? c) If you can play repeatedly, staking a Kelly fraction $f$ of wealth against a fixed ticket price $p$ per play, derive the condition on $p$ and $f$ for positive expected log-growth and give the optimal $f^*$. d) Discuss how a house cap (maximum payout $C$) affects the expected value and your reservation price.

Overview: This question tests probability distributions, divergent expected values, utility theory, and optimal betting strategies (Kelly staking) in the Statistics & Math area for a Data Scientist role.

This question is drawn from a data scientist interview experience.