Finding probability

Question

Suppose an airline accepted 12 reservations for a commuter plane with 10 seats. They know that 7 reservations went to regular commuters who will show up for sure. The other 5 passengers will show up with a 50% chance, independently of each other. (a) Find the probability that the flight will be overbooked. (b) Find the probability that there will be empty seats.

Answer 1

You have 5 passengers and 3 empty seats.

a) Pby (overbooked) = Pby (5 show up) + Pby (4 show up) = (.5)^5 + 5(.5)^5 = 6/32 = .1875

b) Pby (empty seats) = Pby (0 show up) + Pby (1 shows up) + Pby (2 show up) = (.5)^5 + 5(.5)^5 + 5C2(.5)^5 = 16(,5)^5 = 16/32 = .5