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

$$

P_n = p(1-P_{n-1}) + (1-p)P_{n-1} \qquad n \ge 1

$$

and use this to prove (by induction) that

$$

P_n = \frac{1+(1-2p)^n}{2}

$$