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### Problem Summary:
An airline has 168 seats available for a flight from Calgary (YYC) to Los Angeles (LAX). The airline books 180 passengers, and 14% of passengers with reservations do not show up. The questions involve calculating expectations, probabilities, and outcomes of overbooking.

We'll tackle each sub-question based on the information given.

### (a) Expected number of people showing up and the standard deviation:

We are given:
- Total passengers booked = 180
- Probability of not showing up = 0.14
- Probability of showing up = 1 - 0.14 = 0.86

The number of people who show up can be modeled by a **binomial distribution** where the number of trials \(n = 180\) and the probability of success (showing up) \(p = 0.86\).

1. **Expected number of people showing up (mean):**
   \[
   \mu = n \times p = 180 \times 0.86 = 154.80
   \]
   So, the airline expects about **154.80 people** to show up.

2. **Standard deviation:**
   The standard deviation for a binomial distribution is calculated using the formula:
   \[
   \sigma = \sqrt{n \times p \times (1 - p)}
   \]
   Substituting the values:
   \[
   \sigma = \sqrt{180 \times 0.86 \times 0.14} = \sqrt{21.672} \approx 4.655
   \]
   Therefore, the standard deviation is approximately **4.66**.

### (b) Probability that 6 seats are empty:

If 6 seats are empty, this means that **162 passengers showed up** (since the plane has 168 seats). We want to find \(P(X = 162)\), where \(X\) is the binomial random variable representing the number of people who show up.

Using the binomial probability formula:
\[
P(X = k) = \binom{n}{k} \cdot p^k \cdot (1 - p)^{n - k}
\]
where:
- \(n = 180\),
- \(k = 162\),
- \(p = 0.86\).

First, calculate the binomial coefficient:
\[
\binom{180}{162} = \frac{180!}{162!(180-162)!}
\]
Then plug into the full binomial expression to calculate the probability. This can be more easily done using a calculator or statistical software.

### (c) Probability a passenger will not make it due to overbooking:

Since there are 180 passengers and 168 seats, there is an overbooking of **12 passengers**. The probability that more than 168 passengers show up is the probability that at least 169 passengers show up.

We need to calculate:
\[
P(X \geq 169)
\]
where \(X \sim \text{Binomial}(180, 0.86)\).

This would typically require summing the binomial probabilities from 169 to 180, which can also be efficiently done using a cumulative binomial probability function from statistical software.

---

Would you like more detailed step-by-step calculations for parts (b) and (c), or any clarifications? 

Here are some further questions for practice:

1. What would be the probability of 10 seats being empty?
2. If the airline booked 190 passengers instead of 180, how would the expectations change?
3. What is the probability that exactly 168 passengers show up?
4. How would increasing the show-up probability to 90% affect the standard deviation?
5. What is the expected number of empty seats if 154 passengers show up?

**Tip**: For binomial probabilities, tools like online calculators, Excel, or Python can speed up computations involving large binomial coefficients.

An airline has 168 seats available for a flight from YYC (Calgary International Airport) to LAX. The airline books 180 passengers, with 14% of passengers not showing up. The following questions are posed:
(a) How many passengers are expected to show up? Calculate the expected number and standard deviation.
(b) What is the probability that exactly 6 seats will be empty?
(c) What is the chance that a passenger will not make it to LAX due to overbooking?

This problem involves the probability of overbooking on a flight from Calgary to Los Angeles, where 180 passengers are booked for 168 seats. The analysis covers expected values, standard deviations, and binomial probability calculations. Suitable for advanced high school or college students.

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