Question

I operate 47 flights a day on 30 seat planes, and for each flight I can sell up to 32 seats at $200 each. If more people show up than I have seats for, I have to pay them $500 each. If I sell 32 tickets, the probability that all 32 show up is 5%, 31 people show up is 10%, and thus the probability that 30 or fewer people show up is 85%. If I sell 31 tickets, the probability that exactly 31 people show up is 10%. If I sell the profit maximizing number of tickets on each flight, how much money would I expect to earn each day? Assume all of the tickets I offer for sale will be bought. Answer to the nearest thousands of dollars without the dollar sign. For example, if your answer is $434.332, enter 434,332 in the box.

Answer 1

We can expect to earn $14,448,000 each day. The expected daily earnings, we need to consider the number of seats available

The probability of different numbers of people showing up, and the amount earned per seat sold.

Let's break down the given information:

- We operate 47 flights a day, with each plane having 30 seats.
- For each flight, we can sell up to 32 seats at $200 each.
- If more people show up than the available seats, we have to pay them $500 each.
- The probability of all 32 people showing up when 32 tickets are sold is 5%.
- The probability of 31 people showing up when 32 tickets are sold is 10%.
- The probability of 30 or fewer people showing up when 32 tickets are sold is 85%.
- The probability of exactly 31 people showing up when 31 tickets are sold is 10%.
Maximize profit, we want to sell the optimal number of tickets on each flight. Since the probability of 30 or fewer people showing up is 85% when 32 tickets are sold, it would be ideal to sell 32 tickets on each flight.

Now, let's calculate the expected daily earnings:
Expected earnings = (Number of flights) x (Number of tickets sold) x (Earnings per seat sold)
Since we operate 47 flights a day, and we want to sell 32 tickets on each flight, the total number of tickets sold in a day would be:

Total tickets sold = (Number of flights) x (Number of tickets sold per flight)
Total tickets sold = 47 x 32
Total tickets sold = 1,504

Therefore, the expected daily earnings would be:

Expected earnings = (Number of flights) x (Number of tickets sold) x (Earnings per seat sold)
Expected earnings = 47 x 1,504 x $200
Expected earnings = $14,448,000

Hence, we can expect to earn $14,448,000 each day.