The Law of Large Numbers for Sports Betting

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.

Let’s take a look at a coin flip with 2.10 in odds of heads. The probability is 50 %, so the edge is 5 % (2.10/2.00 = 1.05). With a stake of $10 and potential return of $11, one trial give an expected value of $0.5 ($11*0.5 - $10*0.5 = $0.5). However, you won’t have $0.5 in profits after the first toss. You will either be $11 up or $10 down. To see how this looks, we can plot the different outcomes and their corresponding probability:

 Results of a coin toss 

Results of a coin toss 

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.

When we flip the coin a second time, there are now four outcomes: $22 up (25 % probability), $1 up (50 %) and $20 down (25 %). The expected value can be calculated to be $1  (2 tosses * $0.5), and with more possible outcomes, the probability distribution would now look like this:

 Expected value of a coin toss

Expected value of a coin toss

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.

Therefore, let’s see what happens if we keep increasing the sample size. Naturally, as the number of possible outcomes increases, winning every bet will become highly unlikely (prob = (1/2)^sample size)) and similarly it will become highly unlikely to lose all of the bets. If we map out the different outcomes and their probability after 1000 trades, we get the following bell-shaped curve:

 Distribution of 1000 sports trades

Distribution of 1000 sports trades

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.

Furthermore, the chances of negative returns is a lot less than before. In fact, we can calculate the probability of losing money after 1000 trades to be 7 % (You’ll have to trust us on this one). We can also plot the same probability for a number of sample sizes, which will turn out like this:

 Probability of being profitable after X number of sports trades

Probability of being profitable after X number of sports trades

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.