The Law of Small Numbers in Sports Betting Explained

April 21, 2017

The human mind has a funny way of playing tricks with people and forcing them into believing that certain things – and particularly outcomes when it comes to betting on sports– are more likely to happen than others.

The law of small numbers, therefore, can be of particular importance for sports bettors as it represents a wrongly conceived view on true probability which states that a relatively small number of observations will reflect the general population.

It basically represents a cognitive bias which stands in direct conflict with the theory of large numbers which is closely connected to gambler’s fallacy as a concept which has people believed that – in the case of a coin flip – the tenth toss on nine consecutive heads is bound to be tails. Contrary to this universally accepted view, the coin has no memory and the probability of landing heads or tails remains the same – 50% as it corresponds to the law of large numbers which determines that deviation from 50/50 chances decrease to the neglected amount in large sums.

The law of small number can have a similar impact on a punter’s mind however and understanding the full concept behind it can help bettors avoid falling into its trap.

The best example to illustrate the law of small numbers has been the so-called Hospital Quiz, which references a study performed by two psychologists in 1974. In an experiment they created, there was a scenario set-up of two towns being served by two hospitals – a small one and a big one.

According to the quiz, 45 babies were born each day in the larger facility, whereas around 15 babies were born daily in the smaller unit. When it comes to determining the sex of the babies, the law of large number and that coin toss example comes into play, stating that 50% of all babies will be boys and the other 50% will be girls.

The number, however playing tricks with the gambler's fallacy, will vary on daily basis, resulting in a larger number of boys one day and a larger number of girls the other day. Within a year’s timeframe, each hospital would record period where more than 60% of the babies would be boys and the question two phycologists asked was – which hospital recorded more such days and periods, the larger hospital or the smaller one?

The wrong answer is the larger hospital, and yet that was the answer 78% of the quiz respondents would give to the question. The number of days when boys were outnumbering girls was nearly three times greater in the smaller hospital.

It is simple, the law of large numbers tells us that larger sample sizes are less likely to let outcome stray from the 50%-mark, compared to a smaller sample size. The two psychologists went on to describe this error in judgment as a belief that the patterns can be formed by small numbers, whereas in fact, they represent an inappropriate perception of the wider population.

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