Question

A box contains 5 green balls. We choose balls at random, with replacement, according to the following rules: (i) Upon choosing each ball from the box, we mark it with a red stripe before replacing it in the box. (ii) We stop as soon as we choose a ball with a red stripe (i.e. the ball has been chosen twice). Let x= the number of times that balls were chosen from the box. (Note that x must be at least 2.) Find the distribution of x.

Answer 1

x is a discrete random variable with mass points 2,3,4.....

P(x=2) = P(a ball with a red stripe was chosen on the 2nd choice) = 1/5 = 0.2

P(x=3) = P(a ball with a red stripe was chosen on the 3rd choice) = (4/5)(1/5) = 0.16

P(x=4) = P(a ball with a red stripe was chosen on the 4th choice) = (4/5)²(1/5) = 0.128

P(x=5) = P(a ball with a red stripe was chosen on the 5th choice) = (4/5)³ 1/5 = 0.1024

P(x=6) = P(a ball with a red stripe was chosen on the 5th choice) = (4/5)⁴ 1/5 = 0.08192

Similarly ; P(x=n) = P(a ball with a red stripe was chosen on the n-th choice) = (4/5)^n-2(1/5)

Thus ; x has a geometric distribution with pmf :-  f_x( j ) = P(X = j) = (4/5) ^j-2(1/5)