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# AP Stats problem Overbooking an Airplane?

Here is the problem, i need to show how i am doing trials to test out the data:

Delta Airlines has a 15-seat commuter turboprop that is used for short flights. Their data suggest that on average about 8% of customers who buy tickets are no-shows. Wanting to avoid empty seats (they see this as missed opportunity to increase revenue), they decide to sell 17 tickets for each flight. Ticketed customers who can’t be seated on the plane will be accommodated on another flight and will receive a certificate for free flight at another time that can be used for any available seat. You have been retained as a consultant to Delta. Your job is to determine if this particular overbooking strategy is sound. Use simulation methods to perform your analysis.

Assign digits to represent outcomes

Simulate many reputations

### 2 Answers

- LeltosLv 51 decade agoFavorite Answer
p = .92

n = 17

Average = .92 * 17 = 15.64

let X be the number of people who show

P(X = 17) = .92^17 =~ 0.2423

P(X = 16) = .92^16 * 0.08 * 17 = 0.3582

Extra Cost of (X = 17) = 2 flights

Extra Cost of (X = 16) = 1 flight

Average extra cost:

2 * 0.2423 = 0.4846

1 * 0.3582 = 0.3582

SUM = 0.8429 flights

17 - 0.8429 = 16.1571

This is more profitable then if they had only 16 or 15 tickets!

P(X = 15) = .92^15 * 0.08^2 * 17*16/2 = 0.2492

P(X < 15) = 1 - 0.2492 - 0.3582 - 0.2423 = 0.245

The odds that people who miss their flight due to overbooking would be able to get a seat on the next flight is lower than the odds that there will be an empty seat next flight!!

The average number of people (15.64) is simply too high.

====

If they sold 18 tickets

P(X = 18) = .92^18 =~ 0.2229

P(X = 17) = .92^17 * 0.08 *18 =~ 0.3489

P(X = 16) = .92^16 * 0.08^2 * 18 * 17 / 2 = 0.2579

Total cost = 3 * 0.2229 + 2 * 0.3489 + 0.2579 = 1.6246

18 - 1.6426 = 16.375

They would make even more money if they booked 18 seats for each flight, but this effect is lessened due to the fact of rollover, eg the passengers would be rolled over to the next flight, thus creating more free tickets etc.. It would become even harder to accomadate the extra passengers.

- dysartLv 44 years ago
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