A coin with probability p of coming up heads is flipped n + k times. Let Rk denote the event that a run of k consecutive heads occurs at least once. Let X1 be the indicator variable of the event that flips 1, . . . , k all land heads, and for i = 2, . . . , n+1, let Xi be the indicator variable of the event that flip i −1 lands tails and flips i, . . . , i +k − 1 all land heads. With W =n+1 i=1 Xi show that

(a) P(Rk) = P(W >0).

(b) Approximate P(W >0).

(c) Bound the error of the approximation.