A fair coin is flipped until a head appears. Let the number offlips required be denoted N (the head appears on the ,\1th flip).Assu1ne the flips are independent. Let the o utcon1es be denoted byk fork= 1,2,3, . ... The event {N = k} 1neans exactly k flips arerequired. The event {,v;;, k} n1eans at least k flips arerequired.
a. How n1any o utcon1es are there?
b. What is Pr[N = k] (i.e., the probability of a sequence of k -1 tails followed by a heads)? (Hint: write a gene ral expressionfor Pr[N = k] for any k = 1,2,3, .. . )
c. Show the probabilities sum to l (i.e., I:f: 1 Pr[,v = k] =1).
d. What is Pr [ N ;;, I] for all I;;: l?
e. What is Pr[N s /] for all I;;: l?
f. Do the answers to tl1e previous two parts sum to l? Shouldthey?
