Question

Each time a fisherman casts his line, a fish is caught with probability p, independent of whether a fish is caught on any other cast of the line. The fisherman will fish all day until a fish is caught and then he will quit and go home. Let Ci denote the event that on cast i the fisherman catches a fish. Draw the tree for this experiment and find P[C1], P[C2], and P[Cn] as functions of p.

DO NOT POST SOMEONE ELSE'S ANSWER. PLEASE SHOW FULL, COMPLETE ANSWER. THANK YOU!

Answer 1

It is given in the question that

Fish is caught with probability p

a fish is caught with probability p, independent of whether a fish is caught on any other cast of the line

and Ci denote the event that on cast i the fisherman catches the fish

Our event will end when Fish will be caught

Let us draw the Diagram

From the Tree

P(Probability that fish will be caught on 1st cast)=

Using the sum of probability law

Probability that fish will not be caught on 1st cast=

Because it is given that both the events are independent then

We will apply the rule of probability of independence

P(Fish caught on 2nd cast) = P(did not caught on 1st event )*P(fish will be caught on 2nd attempt)

Because it is given that a fish is caught with probability p, independent of whether a fish is caught on any other cast of the line

So probability that fish will be caught on 2nd attempt will be also p

substitute the values

Let us consider the last case

Finally fish will be caught on the nth cast if fish will not be caught on previous (n-1) casts

Thus