Each time a fisherman casts his line a fish is caught with p

Each time a fisherman casts his line, a fish is caught with probability p, independent of whether a fish is caught or any other cast of a line. Assume that the fisherman will fish all day until a fish is caught and then he quits and goes home. What is the probability that the fisherman goes home after the first cast? the second cast? and the nth cast? Assume that the fisherman casts Ins line exactly n times per day then he quits and goes home. What is the probability that the fisherman catches during Ins day no fishes? 5 fishes and n fishes?

Solution

Prob for catching a fish = p

X- no of fish caught is binomial since there are two outcomes and each is independent of the other

i) Prob he goes home after I cast = P(X=1) = p

ii) Prob he goes home after II cast = Not getting fish in first and getting fish in second

= (1-p) p

iii) Prob he goes home after nth cast = (1-p)^n-1p

3.2) i) Prob no fishes = P(X=0)

= (1-p)ⁿ

ii) Prob (X=5) = nC5 p⁵(1-p)^n-5

iii) P(X=n) = pⁿ