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Let's break down the problem step by step.

### Given:
- The airline has **167 seats** available.
- The airline has **181 passengers** booked for the flight.
- **12%** of passengers do not show up for the flight.

### Part (a): 
We need to calculate the **expected number of passengers** that will show up and the **standard deviation**.

The problem suggests that each passenger shows up independently with a certain probability, which implies a **binomial distribution**.

In a binomial distribution:
- \( n = 181 \) (number of passengers)
- \( p = 1 - 0.12 = 0.88 \) (probability of showing up)

#### Expected Value:
The expected number of people showing up is given by:
\[
E(X) = n \times p = 181 \times 0.88 = 159.28
\]

#### Standard Deviation:
The standard deviation for a binomial distribution is given by:
\[
\sigma = \sqrt{n \times p \times (1 - p)} = \sqrt{181 \times 0.88 \times 0.12} = \sqrt{19.0368} \approx 4.36
\]

### Part (b):
We need to calculate the probability that there will be **7 empty seats**, which means that **160 passengers** show up. The number of passengers showing up follows a binomial distribution \( X \sim B(n=181, p=0.88) \).

We are asked to find \( P(X = 160) \). The formula for the binomial probability mass function is:
\[
P(X = k) = \binom{n}{k} p^k (1 - p)^{n-k}
\]
where \( \binom{n}{k} \) is the binomial coefficient.

Let me calculate this next.The probability that exactly 160 passengers show up (i.e., 7 seats are empty) is approximately **0.0910**, or 9.10%.

### Summary:
- The expected number of people showing up for the flight is **159.28**.
- The standard deviation of the number of people showing up is **4.36**.
- The probability that exactly 7 seats will be empty is approximately **0.0910** (or 9.10%).

Do you have any questions or need further details? Here are five related questions to explore:

1. How would the probability change if the no-show rate increased to 15%?
2. What is the probability that more than 167 passengers show up?
3. How would overbooking affect the airline's operational strategy?
4. What would be the expected number of people showing up if 190 passengers were booked?
5. How could the airline use these probabilities to optimize the number of seats sold?

**Tip:** Binomial distributions are useful when modeling situations with independent trials, such as the probability of each individual passenger showing up.

A certain airline has 167 seats available for a flight from YYC to LAX. The airline books 181 passengers, and 12% do not show up for their flight. Calculate (a) the expected number of people to show up, and the standard deviation, and (b) the probability that there will be 7 empty seats.

Learn how to calculate the expected number of passengers who will show up for a flight, as well as the standard deviation, given overbooked flights with no-show probabilities. Additionally, calculate the probability of having exactly 7 empty seats. Ideal for college-level statistics students.

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