Question

Suppose that for a given city, a grocery store has a 30% chance of having toilet paper in stock. Let X represent the number of grocery stores we must visit until we find the first one that has toilet paper in stock. a) What is the probability distribution of X? b) what is the probability that we must search 2 grocery stores until we find the first one that has toilet paper? c) what is the probability that we must search 5 or more grocery stores until we find the first one that has toilet paper? d) find the mean and standard deviation of X.

Answer 1

Part a)

X ~  Geomteric (P = 0.3 , q = ( 1 - P ) = 0.7 )
P ( X = x ) = P ( 1 - P )( x - 1 )

Part b)

Part c)

P ( X = 1 ) = 0.3
P ( X = 2 ) = 0.21
P ( X = 3 ) = 0.147
P ( X = 4 ) = 0.1029

P ( X >= 5 ) = 1 - P ( X <= 4 ) = 1 - 0.7599 = 0.2401

Part d)

Mean = 1 / P = 1 / 0.3 = 3.3333
Variance = q / P2 = 0.7 / 0.09 = 7.7778
Standard deviation = √(7.777778) = 2.7889