https://file.aiquickdraw.com/mathbot/file/1729064334474ky4huhh9.jpg

The problem you've provided is based on the scenario of overbooked flights, where the airline expects a certain proportion of people to not show up. This can be modeled using the binomial distribution, given that each passenger independently shows up for the flight with a certain probability. Let's go through it step by step.

### Given Information:
- The airline has **165 seats** available on the flight.
- **175 passengers** have booked tickets.
- The probability of a person **not showing up** is **11%**. Thus, the probability of a person **showing up** is:
  \[
  p = 1 - 0.11 = 0.89
  \]

### Part (a) - Expected Number and Standard Deviation of People Showing Up:
We can treat this as a binomial problem with the following parameters:
- **n** = 175 (total passengers)
- **p** = 0.89 (probability of a passenger showing up)

The **expected number of people** showing up for the flight is given by:
\[
E(X) = n \cdot p = 175 \cdot 0.89 = 155.75
\]

The **standard deviation** is given by the formula for binomial distribution:
\[
\sigma = \sqrt{n \cdot p \cdot (1 - p)} = \sqrt{175 \cdot 0.89 \cdot 0.11} = \sqrt{17.1575} \approx 4.14
\]

So, the answers for part (a) are:
- **Expected number of people**: 155.75
- **Standard deviation**: 4.14

### Part (b) - Probability that 6 Seats are Empty:
We want to calculate the probability that **6 seats are empty**, meaning that exactly 159 passengers show up (since 165 - 6 = 159).

We need to calculate the binomial probability:
\[
P(X = 159) = \binom{175}{159} \cdot (0.89)^{159} \cdot (0.11)^{16}
\]
This can be computed using statistical software or a calculator capable of handling binomial probabilities.

For the sake of time, I'll compute this using Python.

### Part (c) - Chance of Overbooking:
The flight has 165 seats. If more than 165 people show up, the flight will be overbooked. So we need to calculate the probability that more than 165 people show up:
\[
P(X > 165) = 1 - P(X \leq 165)
\]
This can also be computed using a cumulative binomial distribution function.

I'll compute both parts (b) and (c) and share the results shortly.### Results:

- **Part (b)**: The probability that exactly 6 seats are empty (i.e., 159 passengers show up) is approximately **0.0752** or **7.52%**.
  
- **Part (c)**: The probability that the flight is overbooked (i.e., more than 165 passengers show up) is approximately **0.0054** or **0.54%**.

If you'd like more details or have any questions, feel free to ask!

Here are 5 related questions to explore further:
1. What happens to the probability of overbooking if more passengers book tickets?
2. How would the results change if the no-show rate was 20% instead of 11%?
3. What is the probability that exactly 165 passengers show up for the flight?
4. How does the binomial distribution change if the flight involves multiple stopovers?
5. Can this binomial model be approximated by a normal distribution for large flights?

**Tip**: When working with binomial probabilities for large \( n \), consider using the normal approximation for quicker calculations!

A certain airline has 165 seats available for a flight from YYC (Calgary International Airport) to LAX (Los Angeles International Airport). Because people with reservations do not show up for their flight 11% of the time, the airline always overbooks this flight. That is, there are more passengers that have tickets on the flight than there are seats. Suppose the airline has 175 passengers booked for 165 seats. Assume one person showing up for the flight does not affect others who may, or may not, show up for this flight. (a) How many people (with tickets) does the airline expect to show up for this flight? Provide the standard deviation as well. Enter your answers to two decimals. (b) When the flight takes off from YYC, what is the probability that there will be 6 seats empty? Enter your answer to four decimals. (c) What is the chance that a passenger with a flight reservation will not make it to LAX due to overbooking? Use four decimals in your answer.

Solve a probability problem involving airline overbooking using binomial distribution. Learn how to calculate the expected number of passengers showing up, the standard deviation, and the probability of having empty seats or an overbooked flight. This problem is perfect for students learning about binomial probability distributions at the undergraduate level.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation