Heads or Tails and the Theory of Probability
A Professional Gambler.... "Whenever you speculate you should aim to get fair value. It is helpful to have an understanding of the mathematics and possibilities involved. Take the spin of a coin. It must be assumed that the coin is evenly balanced and fairly spun. The chances of it landing head or tail uppermost are equal.
In any series of events where the outcome of each event is theoretically even, such as head or tail in the spin of a coin or red or black in roulette (ignoring zero), it is called equipartion when each alternative comes up an exactly equal number of times.
An example would be spinning a coin twelve times, resulting in six heads and six tails. It is widely believed that according to the Law of Averages the longer the series of events where only two equally likely results can occur the closer will become equipartion. This suggests that because "in the long run" numbers will even out a run of tails, say, will be followed by a run of heads.
There is no justification for this. With an equally balanced coin and fair spins if head comes up a dozen times in a row the chance of the 13th spin being heads or tails is exactly even. The coin knows nothing about the previous spins. I walk up to a roulette table aiming to have a fiver on red or black. An onlooker tells me that black has won the last nine spins. Certainly I will be inclined to select red thinking that after nine consecutive blacks a change is due, yet, I know this is a fallacy. Red might come up of course but there is an exactly equal chance of another black.
Perhaps the confusion arises because one thinks, after hearing of the nine consecutive blacks, the odds must be very great against ten blacks in a row so I will go for red. Yes indeed, the odds against ten consecutive blacks are great but only before the ten spins start. On the single tenth spin black is still exactly an even chance.
Do not assume, after reading the above observations, that I reject the "Law of averages" or what 1 prefer to call the "Theory of probabilities". It is correct to expect that in a long series of events, provided all the conditions remain constant, results will tend to average out in relation to the actual odds and chances. The longer the run the lesser, indeed, will be the deviation from the mathematically expected outcome but in percentage terms not in numbers.
Where the number of possible results is two the change of equipartion worsens as the number of events increases. The larger the number of events the smaller will become the deviation from normal in percentage terms but larger in numbers.
Remember - in matters of pure chance such as the tossing of a coin, the rolling of dice, the spinning of a roulette wheel - what has happened in the past has no bearing on the result of the current event.Remember - in matters of pure chance such as the tossing of a coin, the rolling of dice, the spinning of a roulette wheel - what has happened in the past has no bearing on the result of the current event."
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