An airline always overbooks if possible. A particular plane has 100 seats on a flight in...

An airline always overbooks if possible. A particular plane has 100 seats on a flight in which a ticket sells for $325. The airline sells 105 such tickets for the flight.

(a) If the probability of an individual not showing up is 0.065, assuming independence, what is the probability that the airline can accommodate all the passengers who do show up? Show your work.

(b) If the airline must return the $325 ticket price plus a penalty of $410 to each passenger that cannot get on the flight, what is the expected payout (penalty plus ticket refund) that the airline will pay? Show your work.

Solutions

Expert Solution

Since the passengers do not show up independently, hence the outcome that a certain number of passengers show up are Bernoulli trials, where 105 "trials" are run and the probability of success that a particular passenger does not show up is 0.065. Thus, parameters of the Binomial Distribution are

(a) The airline will be able to accomodate all passengers, if the number of passengers that do not show up is at least 5. Since in such a scenario. there will be less than or equal to 100 passengers, and the airline does have 100 seats. Hence, required probability is

(b) The refund amount for a single extra passenger that cannot be accomodated is

The airline will need te refund a certain no of passengers, only if the Random Variable X in above distribution takes values from 0 to 4, inclusive. For x=0, the airline needs to refund 5 passengers. For x=1, it will be 4 passengers, and so on till x=4, it will be for 1 passenger. In order to compute the expected payout, we should multiply the number of passengers times the refund amount, with the Binomial probability for those many passengers to not show up. Hence, the total expected refund amount can be written as

We can best compute this value using BINOM.DIST in excel, and then running the summation as

x	P(X=x)	Expected Payout
0	0.000861	3.165749
1	0.006288	18.48662
2	0.022731	50.12148
3	0.054254	79.75361
4	0.096178	70.6907
		222.2181

So we obtain the expected payout value as $222.22 to nearest cent.

orchestra answered 1 year ago

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