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Why does a fair coin having a one-half probability of heads not predict the next toss?

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sample space for a pair of dice — probability theory Sample space for a pair of dice.

In probability theory, a fair coin having a one-half probability of heads does not predict the next toss because each coin toss is a stochastically independent event.12

Independence means that the outcome of any single trial is not influenced by the results of previous tosses. While the law of large numbers dictates that the ratio of heads to total tosses will approach 1/2 over a long series of events, this is a statistical property of the aggregate, not a mechanism that forces individual outcomes to balance out.123

The belief that past results influence future outcomes in games of pure chance is known as the doctrine of the maturity of the chances (or the gambler's fallacy). This is a misconception; the coin has no "memory," and the probability of heads remains exactly 1/2 for every individual toss, regardless of whether the previous results were heads or tails.2