Question

An airline finds that with probability 0.04 every passenger that makes a reservation on a particular flight will not show up. Consequently, their policy is to sell 200 reserved seats on a plane that has only 196 seats. (a) Find the probability that every person who shows up for the flight will find a seat available. (b) Use the Poisson approximation to compute this probability and compare your answer to the exact answer that you found in part (a).

Answer 1

a)

probability that every person who shows up for the flight will find a seat available

=P(at least 4 people not show up)=1-P(at most 3 people not show up)

=1-P(X=0)-P(X=1)-P(X=2)-P(X=3)

=1-²⁰⁰C₀(0.04)⁰(0.96)²⁰⁰ -²⁰⁰C₁(0.04)¹(0.96)¹⁹⁹ -²⁰⁰C₂(0.04)²(0.96)¹⁹⁸ -²⁰⁰C₃(0.04)³(0.96)¹⁹⁷

=1-0.0395=0.9605

b)

from poisson distribution ; expected number of people not arrive =np=200*0.04=8

hence from Poisson approximation; probability that every person who shows up for the flight will find a seat available =1-P(X=0)-P(X=1)-P(X=2)-P(X=3)

=1-e^-8*8⁰/0!-e^-8*8¹/1!-e^-8*8²/2!-e^-8*8³/3! =1-0.0424=0.9576