The image shows a probability and statistics problem related to airline overbooking. Here's a breakdown of the problem and the answers provided:

Problem Context:

• A flight from Calgary (YYC) to LAX has 165 seats available.
• The airline overbooks the flight because 11% of people with reservations do not show up.
• In this case, 180 passengers are booked for the flight with 165 seats.

Parts of the problem:

(a) How many people does the airline expect to show up, and what is the standard deviation of the number of people showing up?

• The expectation is calculated as $E(X) = np$, where:

  • $n = 180$ (the total number of passengers booked),
  • $p = 0.89$ (since 11% don’t show up, 89% show up).

  The expectation is given as 160.20.

• The standard deviation is $\sigma = \sqrt{np(1-p)}$, which is approximately 4.1978.

(b) The probability that there will be exactly 7 empty seats when the flight takes off:

• This involves a binomial distribution calculation. The answer provided is 0.0791.

(c) The chance that a passenger with a flight reservation will not make it to LAX due to overbooking:

• The provided answer is 0.8541.

Would you like a detailed explanation on how these probabilities and calculations were derived, or do you have any further questions?

5 Follow-up Questions:

1. How is the binomial distribution applied to solve overbooking problems like this?
2. How does the concept of standard deviation help in understanding the spread of outcomes for passengers showing up?
3. What is the significance of calculating the exact number of empty seats in overbooking models?
4. How would the probabilities change if the no-show percentage were different, say 5% instead of 11%?
5. What real-world assumptions are made in using the binomial distribution for this type of problem?

Tip:

In probability questions involving binomial distribution, always verify whether the event in question has only two possible outcomes (like "show up" vs. "not show up"). This makes binomial modeling suitable!