Answer to Question #116053 in Statistics and Probability for Ulrich

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

(a) Find the probability that the flight will be overbooked.

$P(X=4)+P(X=5)=$ C_5,4(0.5)⁴(1−0.5)⁵⁻⁴+C_5,5(0.5)⁵(1−0.5)⁵⁻⁵=(5+1)(0.5)⁵=0.1875

(b) Find the probability that there will be empty seats.

$P(X=0)+P(X=1)+P(X=2)=$ C_5,0(0.5)⁰(1−0.5)⁵⁻⁰+C_5,1(0.5)¹(1−0.5)⁵⁻¹+C_5,2(0.5)²(1−0.5)⁵⁻²=(1+5+10)(0.5)⁵=0.5