The Law of Averages

The Law of Averages

The law of averages, or correctly named the law of large numbers, is one of the most misunderstood concepts of gambling and mathematics. Punters more often than not use it in an incorrect context.

Many pokie machine players believe that if the machine hasn't paid anything reasonable of late, then it is 'due'. Consequently they continue gambling believing that the next score is getting closer and closer with every spin.

Unfortunately this misunderstanding of the law of averages is the downfall of a lot of gamblers. Tied into the law of averages is the similar theory of regression to the mean. It is understandable why some punters misunderstand what it means.

The theory of the law of large numbers (law of averages) states that a sample average will get closer to a population average the more you sample. It’s often quoted, to make more sense, that that ratio of wins and losses will become closer to what is expected. To show you how this works, let’s look at an example.

Suppose we are flipping a coin. We know that the coin has a 50% chance of landing on heads or tails. But suppose we have already thrown it 50 times and we recorded 35 heads and only 15 tails. Gamblers who misunderstand the law of averages will believe that as there have been more heads in the sample that tails is more likely to lob in the future.

This of course is completely incorrect. Each toss of the coin is independent. In other words, it doesn’t matter how many heads was previously tossed; the probability of a tail is still 50%. The coin isn't aware of what the previous spin was. Similarly with pokie machines; it doesn’t matter if a particular machine has won or lost recently, the return is always going to be negative. The same applies with most casino games. Just because six reds have being spun in a row in roulette, it doesn't mean that it is more or less likely for the seven spin to be a red.

So how do gamblers get this theory completely mixed up? Well let’s go back to our example that we have thrown 35 heads and 15 tails. This means 70% of our tosses have been for heads, which is well above the 50% that it should theoretically average. But according to the theory of large numbers, the more times we toss the coin, the more likely the ratio of heads to tails will get closer to 50%. This does not mean that more tails will be tossed however as most gamblers would conclude.

If we were to toss the coin another 950 times and recorded 500 heads and 450 tails, then the ratio would now be . So even though still more heads have been thrown than tails after the initial 50 throws, the ratio gets closer to 50% as the law of large numbers suggest.

So what has this therefore got to do with gambling? Absolutely nothing. Essentially the law of averages assumes that you know the average to start off with. In other words we know that the probability of a coin landing on heads is 50%. But never in sports gambling do you know the probability over the long term of a reoccurring event! Further to that, sports matches are not independent of each other which the law of averages assumes!

The law of averages has absolutely nothing to do with sports and horse racing gambling.

It does however have practicalities in casino’s because we know the probability of success in reoccurring casino gambling (such as betting on red or black in roulette) it can be applied here. However, sorry to say, that there is no way in which the law of averages can help one gain an advantage over the house and profit over the long term. So to put it frankly, the law of averages for all gamblers should be put away in the back shelf never to be touched again. Just remember that all casino games (with the exception of card games) and pokie machines etc. are independent. It doesn’t matter what happened before, the probability of an event (winning) will not change, and will always be in the casino’s favour.

Related to the theory of the law of averages, but still quite interestingly different is the theory of regression to the mean.

Matt Elliott

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