Question

An airport limousine can accommodate up to four passengers on any one trip. The company will accept a maximum of six reservations for a trip, and a passenger must have a reservation. From previous records, 25% of all those making reservations do not appear for the trip. Answer the following questions, assuming independence wherever appropriate. (Round your answers to three decimal places.)

(a) If six reservations are made, what is the probability that at least one individual with a reservation cannot be accommodated on the trip?

(b) If six reservations are made, what is the expected number of available places when the limousine departs?

(c) Suppose the probability distribution of the number of reservations made is given in the accompanying table. Number of reservations 3 4 5 6 Probability 0.09 0.25 0.29 0.37 Let X denote the number of passengers on a randomly selected trip.

Obtain the probability mass function of X. x 0 1 2 3 4

Answer 1

a)

P(arrival of a random person) =p=			0.75
P(at least one can't)=1-P(at most 4 will appear)=1-∑x=04     (6Cx)px(1−p)(6-x) =						0.5339

b)

expected available spaces =E(4-x) =

∑x=04 (4-x)*(6Cx)px(1-p)6-x=

0.2119

c)

let y be the number of reservations made:
probability mass function of X: =P(X=x)=Σy=36 P(Y=y)*P(X=x|y)=Σy=36 P(y)*(yCx)*px(1-p)y-x

x	0	1	2	3	4
P(x)	0.0028	0.0302	0.1284	0.2687	0.5699