Bernoulli's theorem licenses the claim that it is 'usually right' (high probability in a frequentist sense) that a small interval around the sample frequency will include the true population frequency

The more problematic step in the argument is the final step, which takes us from the claim that samples match their populations with high probability to the claim that having seen a particular sample frequency, the population from which the sample is drawn has frequency close to the sample frequency with high probability. The problem here is a subtle shift in what is meant by “high probability”, which has formed the basis of a common misreading of Bernouilli’s theorem. Hacking (1975: 156–59) put