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, 20% of all those making reservations do not appear for the trip. Answer the following questions, assuming independence wherever appropriate. 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. Let X denote the number of passengers on a randomly selected trip. Obtain the probability mass function of X.

Answer : Step 1 of 3 : Given, 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, 20% of all those making reservations do not appear for the trip a) If six reservations are made, the claim is to find the probability that at least one individual with a reservation cannot be accommodated on the trip Where, p = 0.80 and n = 6 P(at least one individual with a reservation cannot be accommodated on the trip) = P(x 5) = P(x = 5) + P(x = 6) 6 5 65 6 6 66 = ( 5(0.80) (1 0.80) + ( 6(0.80) (1 0.80) = (6) (0.32768) (0.20) + (1) (0.2621) ( 1) = 0.3932 + 0.2621 = 0.6553 Step 2 of 3 : b) x 0 1 2 3 4 5 6 h(x) 4 3 2 1 0 0 0 If six reservations are made, the claim is to find the expected number of available places when the limousine departs E(h(x)) = h(x) P(X = x) 0 =(4) (0.00010 +..............+(0) (2621) = 0.117 0.117 be the expected number of available places when the limousine departs.