A Small Regional Flight Carrier

Question

A small regional carrier accepted 17 reservations for a particular flight with 15 seats. 8 reservations went to regular customers who will arrive for the flight. Each of the remaining passengers will arrive for the flight with a 46% chance, independently of each other.

(Report answers accurate to 4 decimal places.)

Find the probability that overbooking occurs.   

Find the probability that the flight has empty seats. 

Answer 1

Hi Anchal,

I will break down the problem step by step.

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Problem Analysis:

1. Total Reservations: 17
2. Total Seats: 15
3. Regular Customers: 8 (these customers will definitely show up)
4. Probability of Each of the Remaining Passengers Arriving: 46% or 0.46
5. Remaining Passengers: Total Reservations - Regular Customers = 17 - 8 = 9

Calculations:

Probability of Overbooking:

1. Overbooking occurs if more passengers show up than there are seats available.
2. Since 8 regular customers will definitely show up, overbooking will occur if more than 7 of the remaining 9 passengers show up (because 8 + 7 = 15, which is the seat limit).
3. We need to calculate the probability of 8 or 9 of the remaining passengers showing up.
4. This is a binomial probability problem, where n=9n=9 (remaining passengers), p=0.46p=0.46 (probability of each passenger showing up).

The probability mass function (PMF) of a binomial distribution is given by:

P(X=k)=(nk)pk(1−p)n−kP(X=k)=(kn​)pk(1−p)n−k

Where (nk)(kn​) is the binomial coefficient, calculated as n!k!(n−k)!k!(n−k)!n!​.

So, for overbooking:

P(overbooking)=P(X=8)+P(X=9)P(overbooking)=P(X=8)+P(X=9)

Probability of Empty Seats:

1. Empty seats occur if fewer than 15 passengers show up.
2. Since 8 regular customers will definitely show up, empty seats will occur if fewer than 7 of the remaining 9 passengers show up.
3. We need to calculate the probability of 0 to 7 of the remaining passengers showing up.

So, for empty seats:

P(empty seats)=P(X=0)+P(X=1)+P(X=2)+...+P(X=7)P(empty seats)=P(X=0)+P(X=1)+P(X=2)+...+P(X=7)

Calculation of Probabilities:

Let's calculate these probabilities using the binomial PMF formula.

Probability of Overbooking:

To calculate the probability of overbooking, we sum the probabilities of exactly 8 and exactly 9 of the remaining 9 passengers showing up.

1. When 8 Passengers Show Up:
2. Binomial Coefficient: (98)=9!8!(9−8)!(89​)=8!(9−8)!9!​
3. Probability: P(X=8)=(98)×0.468×0.541P(X=8)=(89​)×0.468×0.541
4. When 9 Passengers Show Up:
5. Binomial Coefficient: (99)=9!9!(9−9)!(99​)=9!(9−9)!9!​
6. Probability: P(X=9)=(99)×0.469×0.540P(X=9)=(99​)×0.469×0.540

Summing these probabilities gives us the probability of overbooking.

Probability of Empty Seats:

To calculate the probability of empty seats, we sum the probabilities of 0 to 7 of the remaining 9 passengers showing up. We use the same binomial probability formula for each case and sum the results.

The detailed calculation for each scenario (0 to 7 passengers) involves the same binomial probability

formula, with different values of kk (number of passengers showing up).

Results:

1. Probability of Overbooking: Approximately 0.0107 (or 1.07%)
2. Probability of Empty Seats: Approximately 0.9893 (or 98.93%)

Hope this helps! Please let me know if you would like to discuss further.

Best regards,

Ben