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Independent trials that result in a success with probability p

Independent trials that result in a success with probability p and a failure with probability 1 − p are called Bernoulli trials. Let Pn denote the probability that n Bernoulli trials result in an even number of successes (0 being considered an even number). Show that

Pn = p(1 − Pn−1) + (1 − p)Pn−1 n ≥ 1

and use this formula to prove (by induction) that

Pn = p(1 − Pn−1) + (1 − p)Pn−1 n ≥ 1

and use this formula to prove (by induction) that

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