In this article our computer expert Neale Yardley investigates probabilities and dispels some common misconceptions many punters have about the so-called law of averages.

During one of my recent visits to P.P.M's office in Dee Why I had a number of conversations with my colleagues about staking plans and in particular those suitable for the casino games I wrote about in my recent P.P.M. series.

We were discussing the 'numbers' games like craps and roulette where the player bets on a particular number (namely roulette wheel number or dice total) coming up. It was not surprising that the staking plans being mooted revolved around trying to predict which wheel number or dice total was going to come up next.

An analogous situation I am sure you are all familiar with is lotto where some numbers appear to keep coming up more frequently than others. What players must decide is whether to include in their entry the numbers that have been occurring more often or the ones that seem overdue. Similarly with two-up, if heads have been thrown four times in a row, do you keep backing heads or change to tails?

Without mentioning any names, one of my colleagues thought that if a particular number on the roulette wheel hasn't occurred for some time then the chances of it occurring are improved since it has become overdue. Conversely someone else in the office at the time suggested that if a particular number keeps coming up then it is likely to keep coming up as the wheel may be biased in its favour.

After calmly suggesting that both these views were wrong, you could have sworn a bombshell had hit the office!

"What about the law of averages,"one said. “Surely if an event happens more than it should then eventually it must stop occurring to even things out. For example a loosing run of bets must eventually be followed by a winning run, musn't it?"

Since newcomers to gambling, and racing in particular, continue to be fooled by the so-called law of averages, I decided we should set the record straight once and for all. Hence this article!

Consider the simple event of tossing a coin. Assuming the coin is unbiased, there is a 50% chance it will come up heads and a 50% chance it will come up tails.

So what happens if five heads in a row come up? Does this mean tails are more likely to occur on the next toss of the coin?

If the answer is yes then you could bet with confidence on tails knowing that if even money were being offered about the outcome you would be getting value odds.

The answer does have something to do with the law of averages. Unfortunately, most people apply the law in the wrong way and come up with the mistaken answer that a run of heads must be followed by a run of tails to even things out.

Even the 18th Century French mathematician jean Le Rond d'Alembert believed that if one side of the coin appeared more frequently then the other side would have to start occurring more often to achieve equilibrium. Unfortunately he was wrong.

What the law of averages says is that in the long run the ratio of heads to tails will get closer and closer to 1:1. This does NOT mean that the difference between the number of heads tossed and number of tails tossed becomes smaller.

In case this confuses you, consider the following results of a computer simulation.

TossesHeadsTails DIFF.RATIO

By now I hope you are convinced that the difference between tails and heads does not approach zero-it actually appears to get larger and larger. Hopefully, none of you are going to be prepared to wait for over 20,000 tosses of a coin just on the off chance that the difference between tails and heads n-tight eventually balance out.

Just out of interest I kept running the above coin tossing simulation to see whether the lead gained by heads was ever reversed. It actually was but not until over 100,000 tosses!

While this surprises many people, it can be mathematically proven that it is quite probable for one side of the coin to lead for such a long time.

As already mentioned, the only thing that converges to a point of equilibrium is the ratio of tails to heads.

So what does all this mean for the average punter?

First, you should disregard progression systems than to try and recover loses based on the theory that a run of loses will eventually be followed by a run of wins. In other words, don't increase your bet sizes after a run of outs.

An important point to note here is that the reverse strategy, namely increasing bet sizes after wins in an attempt to recover loses, is also unsound. (The only time you should increase your bet size is after your bank has grown in size.)

If you are still not sure why this is the case then consider the event of tossing a coin. Such an event can happen anywhere at any time and in no way depends upon the outcome of previous tosses. In other words, the event of tossing a coin has no memory of what has happened before. The only thing that determines the probability is the mathematical fact that the coin has two sides and there is therefore an even chance of either side coming up.

So runs of heads are not followed by tails to even things out and nor are they followed by more heads just because things appear to be on a roll.

Perhaps a racing example might help clarify this.

Based upon what has just been said, it favourites have been unplaced in the first four races on the card you cannot assume they will start winning to make up for their earlier lack of success. Conversely you cannot assume that a run of winning favourites is going to continue.

As with the coin example, the outcome of any particular race is in no way influence by previous races. There is no magical influence to make a favourite more likely to win just because favourites haven't won for a while.

It cannot be stressed enough that the probability of an event occurring has nothing whatever to do with whether similar events occurred in the past or how often they might have occurred in the past.

Poker machines provide us with another useful example. The frequency of particular payouts is guaranteed by the number of times certain symbols appear on certain wheels of the machine. Do not for a moment think that a poker machine decides to start paying out more because it has been paying less than average up until a given point in time.

The fact that the gambling events given in the above examples have no memory is a deciding factor as to whether a particular staking plan has any merit or not.

Consider the reverse gambling system as an example. It states that if a particular system (e.g. increasing bets after a loss) does not win then the reverse (e.g. decreasing bets after a loss) must win. Hopefully I have already convinced you that this is rubbish but to make sure, consider the following.

Because races do not have 'memory', the result of race 2 is in no way affected by the result of race 1 and so on. This being the case, you would be no better or worse off if the authorities decided to run race 8 first followed by race 7 and so on through to race 1.

It follows, therefore, that it would make no difference either if you reversed your staking plan.

Readers wishing to know more about the racing services provided by Neale Yardley's company, Capital Information Services, can write to him at G.P.O. Box 2559, Canberra, A.C.T. 2601.


By Neale Yardley