In the marble game in Exercise 44, suppose the probabilities depend on the number of marbles left in the ring, N. Let

P(A) = 1/(N+1) , P(B)= (0.2N)/(N+1) , P(C)= (0.8N)/(N+1)

Exercise 44

In a marble game a shooter may (A) miss, (B) hit one marble out and stick in the ring, or (C) hit one marble out and leave the ring. If B occurs, the shooter shoots again.

(a) If P(A) = p₁P(B) = P₂and P(C) p3 and these probabilities do not change from shot to shot, then express the probability of getting out exactly three marbles on one turn.

(b) What is the probability of getting out exactly x marbles in one turn?

(c) Show that the probability of getting one marble is greater than the probability of getting zero marbles if

p1<(1-P2)/((2-P2)