Bayesian Coin Inference and Kelly Betting with Beta Prior
A coin has an unknown bias $p$. You start with a $\text{Beta}(a, b)$ prior on $p$, then observe $H$ heads and $T$ tails.
1. Compute the posterior distribution of $p$ and the posterior predictive probability that the next single flip is heads.
2. Generalize: what is the posterior predictive probability that exactly $h$ of the next $k$ flips are heads?
3. You are offered an even-money bet: you pay $\
$ to play and receive $\
$ if the next flip matches your call (heads or tails). You want to maximize expected logarithmic utility (Kelly criterion). Which side should you bet on, what fraction of your bankroll should you wager, and what is the expected log growth rate per bet?Open the full interactive solver, hints, and worked solution →