Someone who bets more than Kelly can do better if for a stretch; someone who bets less than Kelly can do better if for a stretch, but in the long run, Kelly always wins. The heuristic proof for the general case proceeds as follows. In a single trial, if you invest the fraction of your capital, if your strategy succeeds, your capital at the end of the trial increases by the factor , and, likewise, if the strategy fails, you end up having your capital decreased by the factor . Thus at the end of trials (with successes and failures), the starting capital of $1 yields Maximizing , and consequently , with respect to leads to the desired result Edward O. Thorp provided a more detailed discussion of this formula for the general case. There, it can be seen that the substitution of for the ratio of the number of "successes" to the number of trials implies that the number of trials must be very large, since is defined as the limit of this ratio as the number of trials goes to infinity. In brief, betting each time will likely maximize the wealth growth rate only in the case where the number of trials is very large, and and are the same for each trial. In practice, this is a matter of playing the same game over and over, where the probability of winning and the payoff odds are always the same. In the heuristic proof above, successes and failures are highly likely only for very large . Multiple outcomes Kelly's criterion may be generalized on gambling on many mutually exclusive outcomes, such as in horse races. Suppose there are several mutually exclusive outcomes. The probability that the -th horse wins the race is , the total amount of bets placed on -th horse is , and where are the pay-off odds. , is the dividend rate where is the track take or tax, is the revenue rate after deduction of the track take when -th horse wins. The fraction of the bettor's funds to bet on -th horse is . Kelly's criterion for gambling with multiple mutually exclusive outcomes gives an algorithm for finding the optimal set of outcomes on which it is reasonable to bet and it gives explicit formula for finding the optimal fractions of bettor's wealth to be bet on the outcomes included in the optimal