Suppose that n independent tosses of a coin having probability p of coming up heads are ma...

Suppose that n independent tosses of a coin having probability p of coming up heads are made. Show that the probability that an even number of heads results is , where q − 1 − p. Do this by proving and then utilizing the identity

where [n/2] is the largest integer less than or equal to n/2. Compare this exercise with Theoretical Exercise of Chapter 3.

Exercise

(a) Prove that if E and F arc mutually exclusive, then

(b) Prove that if Ei,i ≥ 1 are mutually exclusive, then