Question

Let a bowl contain 15 chips of the same size and shape. Only one of those chips is red. Continue to draw chips from the bowl, one at a time at random and without replacement, until the red chip is drawn. Show your work.

a) Find the probability mass function of X, the number of trials needed to draw the red chip.

b) Compute the mean and variance of X.

c) Determine P(X <= 10).

Answer 1

Let X be the random variable denoting the number of trials need to draw the red chip.

The bowl contains 15 chips and only one of those chips is red.

Therefore, the probability of getting a red chip is p = 1/15

a) Answer :

X Geometric (p = 1/15)

The pmf of X is

P(X = x) = p * (1 - p)x-1 ; x =1, 2,.....

              = 0                   ; otherwise

The probability mass function of X is

P(X = x) = (1 / 15) * (14 / 15)x-1 ; x = 1, 2, ....

              = 0                                 ; otherwise

b) Answer :

Mean of X is

E(X) = 1 / p = 1 / (1 / 15) = 15

Variance of X is

V(X) = (1 - p) / p2 = (14 / 15) / (1 / 15)2

        = 210

Mean of X is 15 and Variance of X is 210.

c) Answer :

P(X<=10) = (1 / 15) * (1 + (14/15) + (14/15)2 + ........ + (14/15)9 )

                = (1 / 15) * (1 - (14 / 15)10) / (1 - (14 / 15) .......(using sum of geometric series)

P(X<=10) = 0.4984