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, 50% 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?

.109

(b) If six reservations are made, what is the expected number of available places when the limousine departs?
1.125

(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	0.15	0.25	0.34	0.26

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
p(x)

Answer 1

(a).The probability is

(b)

The available places can be 0, 1, 2, 3, 4. Let Y is a random variable shows the number of seats available. Number of seats available 0 if 2 or less customers will not shows up. So

Number of seats available 1 if 3 customers will not shows up. So

Number of seats available 2 if 4 customers will not shows up. So

Number of seats available 3 if 5 customers will not shows up. So

Number of seats available 4 if 6 customers will not shows up. So

The expected number of available places is

Y	p(y)	y*p(y)
0	0.34375	0
1	0.3125	0.3125
2	0.234375	0.46875
3	0.09375	0.28125
4	0.015625	0.0625
Total	1	1.125

So expected number of available spaces is 1.125.

(c)

Let Y shows the number of reservation made. The X will take zero when no customer will show up. The probability that X=0 and Y=3 will be

So by the law of total probability we have

Likewise

When X=4 is not possible for Y=3. And for Y=4 we need to find the probability that all customer will shows up (that is zero customer will not show up). For Y=5 , we need find that zero or one customer will not show up and for Y=6 we need to find that zero, one or two customers will not show up. So