Using a little math, here’s Kelly’s answer for how much to bet in the case of the coin. Let n, W and L be the number of tosses, wins and losses, respectively. The win and loss probabilities are p and q = 1 – p. We start with bankroll X0 and bet a fixed fraction f at each trial. After n tosses we have Xn=X0 (1+f )W(1-f )L. With a little rearranging we have

Xn/X0=exp{n log ([Xn/X0]1/n) = exp{nGn(f)}} where

Gn(f)=log ([Xn/X0] 1/n) = (W/n) log (1+f) + (L/n) log (1-f)

measures the exponential rate of growth per trial.The expected value of Gn(f) is g(f) = p log(1+f) + q log(1-f) which has a maximum at f* = p – q. In this instance f* happens to be the same as the bettor’s edge, namely the expected value of one unit bet.

(1) If g(f) > 0 then limn→∞ Xn = ∞, almost surely, i.e. one’s fortune tends to infinity with probability one: the zone of positive growth.

(2) If g(f) < 0 then limn→∞ Xn = 0, almost surely, i.e. one’s fortune tends to zero with probability one: the zone of ruinous over betting.

(3) If g(f) = 0 then Xn oscillates wildly from 0 to ∞.

(4) With the Kelly strategy Φ* verus any “essentially different” strategy Φ, the ratio Xn(Φ* )/ Xn(Φ) tends to infinity with probability one.

(5) The expected time to reach any specified goal tends to be least with Kelly.

Warrant Diagram