Question

A small regional carrier accepted 16 reservations for a particular flight with 12 seats. 10 reservations went to regular customers who will arrive for the flight. Each of the remaining passengers will arrive for the flight with a 50% chance, independently of each other. (a) find the probability that overbooking occurs (b) find the probability that the flight has empty seats

Answer 1

Let p = the probability that the passengers will arrive for the flight = 0.50

If more than 2 persons out of 6 persons are arrive then there is occurrence of overbooking.

!0 of the customers are certainly arrive, so there are 6 places of randomness.

So that n = 6

X = number of passengers are arrive randomly

So X takes values as 0, 1, 2, 3, 4, 5, and 6

a) If more than 2 persons out of 6 persons are arrive then there is occurrence of overbooking.

So that we want to find P( X > 2 ) = 1 - P(X <= 2 ) ... ( 1 )

Let's use excel to find the binomial probability

P(X <= 2 ) = "=BINOM.DIST(2,6,0.5,1)" = 0.34375

Plug this value in equation ( 1 ), we get

P( X > 2 ) = 1 - 0.34375 = 0.65625

b) If less than 2 persons are arrive then  that the flight has empty seats.

So here we want to find P( X < 2) = P( X <= 1 ) = "=BINOM.DIST(1,6,0.5,1)" = 0.109375