How much to bet

Idea. Repeated bets multiply your wealth, so the long-run growth rate is the expected logarithm of one multiplier. Maximise it.

Step 1: growth rate. One bet multiplies wealth by $1+f$ with probability $0.6$ and by $1-f$ with probability $0.4$. By the law of large numbers, $\frac{1}{n}\log W_n\to g(f)$ with $$g(f)=0.6\log (1+f)+0.4\log (1-f).$$

Step 2: maximise. $$g^{\prime }(f)=\frac{0.6}{1+f}-\frac{0.4}{1-f}=0$$ means $0.6(1-f)=0.4(1+f)$, that is $f=0.2$. $g$ is concave, so this is the maximum: bet $\frac{1}{5}$ of your wealth each time.

The Kelly formula. For a bet paying $b$ to $1$ with win probability $p$ and $q=1-p$, the optimal fraction is $f^{\ast }=\frac{bp-q}{b}$, the edge divided by the odds. Here $b=1$: $f^{\ast }=p-q=0.2$.

Why not more? Expected wealth is largest if you bet everything, but one tail ruins you. Betting twice the Kelly fraction ($f=0.4$) gives $g(0.4)=0.6\log 1.4+0.4\log 0.6\approx -0.0025<0$: in the long run you almost surely lose money even though every bet has positive expectation.