A certain airline has 167 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 13% 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 179 passengers booked for 167 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.

The expectation of the number of people showing up for the flight

The standard deviation of the number of people who show up for the flight

This problem involves a **binomial distribution**, where each passenger has two possible outcomes: either they show up for the flight or they don't. The binomial distribution is appropriate because each passenger's decision to show up is independent of the others.

We are given:
- The number of passengers booked: \( n = 179 \)
- The probability that a passenger shows up: \( p = 1 - 0.13 = 0.87 \)

The binomial distribution has the following properties:
- The **expected number** of passengers who show up (mean) is given by:
  \[
  E(X) = n \cdot p
  \]
- The **standard deviation** is given by:
  \[
  \sigma = \sqrt{n \cdot p \cdot (1 - p)}
  \]

### Step 1: Calculate the Expected Number of Passengers
\[
E(X) = 179 \cdot 0.87 = 155.73
\]

### Step 2: Calculate the Standard Deviation
\[
\sigma = \sqrt{179 \cdot 0.87 \cdot (1 - 0.87)} = \sqrt{179 \cdot 0.87 \cdot 0.13} = \sqrt{20.2401} \approx 4.5
\]

### Final Answer:
- The **expected number** of passengers who will show up is **155.73**.
- The **standard deviation** of the number of passengers who will show up is approximately **4.5**.

Would you like more details or have any questions?

Here are some related questions to explore:
1. What is the probability that exactly 167 people show up for the flight?
2. How does overbooking affect customer satisfaction and operational efficiency?
3. What if the no-show rate changed to 10%? How would that affect the calculations?
4. How does the binomial distribution simplify assumptions in real-world modeling?
5. What strategies do airlines use to minimize the impact of overbooking?

**Tip:** In binomial distributions, probabilities and outcomes are easier to compute when you know both \( p \) (success probability) and \( 1 - p \) (failure probability).

A certain airline has 167 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 13% 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 179 passengers booked for 167 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.

Calculate the expected number of passengers and the standard deviation for an overbooked flight from Calgary to Los Angeles. Learn how to use the binomial distribution, with 179 passengers booked, a 13% no-show rate, and 167 seats available. Suitable for college-level students in statistics and probability.

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