If an unbiased coin is tossed six times, and each time it comes down tails, it is exactly a 50/50 chance that the seventh time the coin is tossed it will come down as a tail. That is fact, yet there are many people who believe the reverse of this statement to be true. Their belief in these fallacies is based upon what they think is the Law of Averages. They believe that it is a 50/50 chance that a tossed coin will come down heads, and that if you go on tossing the coin often enough you will ultimately end up getting exactly 50% heads and 50% tails. Anyone who believes this, is WRONG.
The word "exactly" makes the contention false. If it read "the longer you go on tossing the coin the nearer you will get to 50% heads" would be a true statement of fact. They proceed to deduce that after 400 tosses of a coin have provided 230 heads the next 400 are likely to provide 230 tails because things have to be evened up in the long run. They are wrong again. The so called "law of averages" says nothing of the sort. In fact there is no such thing as the "law of averages". There is only a Law of Great Numbers. It is of utmost importance that this should be understood.
If you toss a coin 100 times it is an even chance that you will get less than 56.75% of either heads or tails. If you toss the coin 1000 times it is an even chance that you will get less than 52.133% heads or tails. And if you toss the coin a million times the odds are that you will get less than 50.0674% heads or tails. So it is perfectly true that the more trials you make of a chance event the nearer the result approaches to the true theoretical probability. The popular way of mis-stating this fact is to say that the effect of luck diminishes as the number of events increases. Whatever may be meant by "the effect of luck", this popular way of expressing a statistical fact is admisssible only if "the effect of luck" is understood as applying to the percentage result. It is because the public take it as applying to the actural numerical result that their conclusions are so dangerously false.
Let me make the difference clear by returning to the 100, 1000 and 1,000,000 tosses of a coin. In 100 tosses it is evens that you will get no more than 56.75% heads or tails, ie: between 44 and 56 heads. In 1,000 tosses it is evens that you will not get more than 52.133% heads or tails, ie: between 479 and 521 heads. Thus on 100 tosses the chances are that you will be only six out either side of the theoretical 50/50, but on 1,000 tosses the chances are that you may be 21 out either side of the theoretical 50/50. The percentage probable error is smaller, but the actural numerical probable error is larger. If you go on to a million tosses the odds are that you will get a 50.0674% result, ie: only 0.0674% difference from the "true" result. But 0.0674% of a million is 674, so your actural numerical result may be as many as 674 out either way. Put these figures side by side and you will recognise the fallacy in the popular idea of the so called law of averages.
So it is actually true that the more tosses you make the more in number and not percentage you are likely to be out from the theoretical 50/50 figure. That is just the opposite of what many people believe.
What I have just highlighted is the mathematical result of the observed fact that what has gone before has, in matters of chance, no influence what so ever on what may happen afterwards.
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