(upon win you are returned $30, winning you $20), then . As an example, if a gamble has a 60% chance of winning (, ), and the gambler receives 1-to-1 odds on a winning bet (), then to maximize the long-run growth rate of the bankroll, the gambler should bet 20% of the bankroll at each opportunity (). If the gambler has zero edge, i.e. if , then the criterion recommends for the gambler to bet nothing. If the edge is negative () the formula gives a negative result, indicating that the gambler should take the other side of the bet. For example, in American roulette, the bettor is offered an even money payoff () on red, when there are 18 red numbers and 20 non-red numbers on the wheel (). The Kelly bet is , meaning the gambler should bet one-nineteenth of their bankroll that red will not come up. There is no explicit anti-red bet offered with comparable odds in roulette, so the best a Kelly gambler can do is bet nothing. Investment formula A more general form of the Kelly formula allows for partial losses, which is relevant for investments: where: is the fraction of the assets to apply to the security. is the probability that the investment increases in value. is the probability that the investment decreases in value (). is the fraction that is lost in a negative outcome. If the security price falls 10%, then is the fraction that is gained in a positive outcome. If the security price rises 10%, then . Note that the Kelly Criterion is valid only for known outcome probabilities, which is not the case with investments. Risk averse investors should not invest the full Kelly fraction. This formula can result in Kelly fractions higher than 1. In this case, it is theoretically advantageous to use leverage to purchase additional securities on margin. Proof Heuristic proofs of the Kelly criterion are straightforward. The Kelly criterion maximizes the expected value of the logarithm of wealth (the expectation value of a function is given by the sum, over all possible outcomes, of the probability of each particular outcome multiplied by the value of the function in the event of that outcome). We start with 1 unit of wealth and bet a fraction of that wealth on an outcome that occurs with probability and offers odds of . The probability of winning