Question

A small regional carrier accepted 17 reservations for a particular flight with 13 seats. 8 reservations went to regular customers who will arrive for the flight. Each of the remaining passengers will arrive for the flight with a 47% chance, independently of each other. a. Find the probability that overbooking occurs. b. Find the probability that the flight has empty seats

Answer 1

Binomial distribution: P(X) = nCx p^x q^n-x

Number of seats = 13

Number of reservations = 17

Number of confirmed reservations = 8

Number of seats remaining = 13 - 8 = 5

Number of reservations remaining, n = 17 - 8 = 9

P(a passenger will arrive for the flight), p = 0.47

q = 1 - p = 0.53

a) P(overbooking occurs) = P(more than 5 will arrive for the flight)

= P(6) + P(7) + P(8) + P(9)

= 9C6x0.47⁶x0.53³ + 9C7x0.47⁷x0.53² + 9x0.47⁸x0.53 + 0.47⁹

= 0.1348 + 0.0512 + 0.0114 + 0.0011

= 0.1985

b) P(the flight has empty seats) = P(less than 5 will arrive for the flight)

= P(X < 5)

= 1 - [P(5) + P(6) + P(7) + P(8) + P(9)]

= 1 - (9C5x0.47⁵x0.53⁵ + 0.1985)

= 0.6807