This is an interesting question to ask. If you are going to the casino, in most cases, the answer is $0. The odds are stacked against you. But what if the odds are in your favor, or you believe that your own predicted probability of winning differs from that of the bet?
The easy answer would be to bet all your money since your expected value is positive. However, if this is a repeated game, the chances of you losing at some point will likely be high. Thus, betting 100% is typically suboptimal. Thus how much should you bet?
One popular criterion for making bets is the Kelly criterion. It is calculated as the ratio of expected winnings to the winnings you would make if you actually won. Formally this is:
- b is the net odds received on the wager (“b to 1″); that is, you could win $b (and get a return of your $1 wagered) for a $1 bet
- p is the probability of winning
- q is the probability of losing [i.e., (1-p)]
One can simplify the expression as:
When the odds are fair [i.e., b = (1-p)/p], the numerator simplifies to 0 and you should not bet. If the odds are worse then the probability, then Kelly bet is negative indicating that you should take the other side of the bet.
Consider this example from Wikipedia:
- b = 2. The odds are a 1:1 payout
- p = 0.70. You have a 70% chance of winning the bet.
If b=2, then the odds believe that the likelihood of winning is 2:1, which indicates that the expected win probability is 66.67% as 66.67/33.33 = 2. However, the true probability is assumed to be 75%. In this case, one should bet:
which is equal to 0.55 or 65%. In other words, if you are given a bet where the payout is 2:1 but your true odds of winning are 70%, you should spend 55% of your bankroll on this gamble.