A careless study of example 10 often results in quite muddled reasoning, of the following form: I have bet on heads successfully ten times, therefore I should bet on tails next. Explain why this muddled reasoning — which has its own name, the gambler’s fallacy in some circles, anti-chance in others — is muddled.

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Now consider a single coin that we flip many times, and where each flip independent of the other. We set up an event istructure that does not reflect the order in which the flips occur. For example, for two flips, we woukd have We assume that P(hh) = p?; a siample computation using the idea of inde- pendence yields that P({ht, th}) = 2p(1 - p) and P(tt) = (1- p). We can generalise this result, to obtain P(k hoads and n - k taik in nfipo) = ()*-p Example 6.10: The probability of earious freiquencies in repcated coin flips Now consider a single coin that we flip many times, and where each flip independent of the other. We set up an event istructure that does not reflect the order in which the flips occur. For example, for two flips, we woukd have We assume that P(hh) = p?; a siample computation using the idea of inde- pendence yields that P({ht, th}) = 2p(1 - p) and P(tt) = (1- p). We can generalise this result, to obtain P(k hoads and n - k taik in nfipo) = ()*-p Example 6.10: The probability of earious freiquencies in repcated coin flips