If you've read our article on value betting, you've learned how edges occur in sports betting, and that good bets are characterized by a positive expected value. The question remains how to transform your edge into what is our ultimate goal: Long term profits.
The chart shows the two only possible outcomes after the first trial, and that both outcomes are equally likely. In other words, after only one coin toss, there’s a 50 % chance that you’ll win money and a 50 % chance that you’ll lose. Strictly speaking, that is a risky investment.
As illustrated, only one of the outcomes results in negative profits. In fact, the probability of losing money has gone down to 25 %, only half of what it was after the first trial. Then again, having a 25 % chance of losing money is still way out of our comfort zone.
The law of large numbers states that the mean of the results obtained from a large number of trials will get close to its expected value. This means that if we toss the coin many times, we should expect it to show heads and tails approximately the same amount of times.
Now, there are several observations to be made. First, we know that the expected value should be $500 ($0.5 * 1000 tosses). This is also reflected in the probability distribution, which is symmetric around 500. That means that you are equally likely to end up above and below $500.
For example, if you want to be 98 % certain to make a profit, you would need a sample size of almost 1900. Increase the number to 5000, and you are theoretically going to make a profit with a certainty of 99.96 %. Moreover, you will earn more than the expected value of $2500 half of the time.
The lesson is simple. As long as your sample size is small, placing valuable bets is only part of the story and risk is still highly present. By increasing your sample size of +EV bets you will reduce the risk of negative returns and your results will converge towards its expected value.
That is the power of the law of large numbers.