Kelly began by analyzing games with a binary win-lose outcome. The key variables are:

• b: The odds define the amount won for a $1 bet. Odds = 5/1 implies a $5 gain if the bet wins, plus recovery of the $1 capital.
• p: The probability defines the likelihood of a favorable outcome.
• f: The share of the current capital to bet.
• V: The value of the capital as a result of betting.

The Kelly rule aims to maximize the value's growth rate, G, of infinitely-repeated bets:

When W and L are the numbers of wins and losses, then:

We can maximize the rate of growth G by maximizing G with respect to f, as illustrated using sympy as follows: 

from sympy import symbols, solve, log, diff

share, odds, probability = symbols('share odds probability')
Value = probability * log(1 + odds * share) + (1 - probability) * log(1 
        - share)
solve(diff(Value, share), share)

[(odds*probability + probability - 1)/odds]

We arrive at the optimal share of capital to bet: