In one experiment, the probability of event a occurrence is p. in n independent repeated experiments, the probability of event a occurrence for odd number of times is calculated. [1 - (1-2p) ^ 2] / 2

In one experiment, the probability of event a occurrence is p. in n independent repeated experiments, the probability of event a occurrence for odd number of times is calculated. [1 - (1-2p) ^ 2] / 2

In n independent repeated trials, the probability of event a occurring once is C (n, 1) * (1-p) ^ (n-1) * P ^ 1; the probability of event a occurring three times is C (n, 3) * (1-p) ^ (n-3) * P ^ 3; the probability of event a occurring five times is C (n, 5) * (1-p) ^ (N-5) * P ^ 5 They are all even terms in the expansion of binomial [(1-p) + P] ^ n, and their sum is the sum of even terms in the expansion of. The expansion of binomial [(1-p) + P] ^ n... (1) and the expansion of binomial [(1-p) - P] ^ n... (2) all odd terms of them correspond to the same, All even numbers correspond to the opposite, so [(1) - (2)] / 2 is: the probability is {[(1-p) + P] ^ n - [(1-p) - P] ^ n} / 2 = [1 - (1-2p) ^ n] / 2
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