How much of the book on a 55/45 coin?

Question

You can bet even money on a coin that you know lands heads with probability $55\%$ : stake $\$ 1 $on heads and you either win$ $1 $or lose$ $1 $. Your bankroll is$ $10{,}000$, and you can repeat the bet many times.

Maximizing expected profit says bet the whole $\$ 10{,}000$ every time. Why is that wrong, and what fraction should you bet?

Accepted Answer

Why "bet it all" fails. Expected profit per dollar staked is 0.55(+1) + 0.45(-1) = \0.10, positive, so maximizing expected P&L on a single bet says stake everything. But you play repeatedly, and wealth compounds multiplicatively. Bet the whole bankroll and the first tails, which arrives 45\% of the time, takes you to zero, after which no edge can help you. Maximizing the average of a compounding process is the wrong objective; you want to maximize its long-run growth rate, which means maximizing expected log-wealth. Kelly fraction. For an even-money bet (b = 1) with win probability p and loss…

How much of the book on a 55/45 coin?

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