Suppose we have a biased coin with unknown probability of landing heads. We play a game consisting of 100 rounds, in which, every round, we must bet $1 on the outcome, in a 1 to 1 payout. What is the expected value of this game?
My initial thought was to observe the first few flips and conditional on those flips, estimate the posterior distribution using a Beta-Binomial model and appropriately bet. However, we must bet $1 each round from the beginning.
Is there any logical way to approach this problem either from a frequentist or Bayesian framework?