This problem involves a binomial distribution since we are dealing with a fixed number of trials (17 flights) and a fixed probability of success (arriving on time). The probability of a flight arriving on time is given as 80%, which means the probability of a flight arriving late is 20%.

We want to find the probability that at least 4 flights arrive late. This can be written as $P(X \geq 4)$, where $X$ is the number of late flights, which follows a binomial distribution:

$X \sim \text{Binomial}(n = 17, p = 0.2)$

The binomial probability mass function is:

$P(X = k) = \binom{n}{k} p^k (1 - p)^{n - k}$

Where:

• $n = 17$ (number of flights),
• $p = 0.2$ (probability of a flight arriving late),
• $k$ is the number of late flights.

To find $P(X \geq 4)$, we can use the complement rule:

$P(X \geq 4) = 1 - P(X \leq 3)$

So, we need to calculate $P(X = 0)$, $P(X = 1)$, $P(X = 2)$, and $P(X = 3)$, and then subtract their sum from 1.

Step-by-Step Calculation:

1. Calculate individual probabilities for $X = 0$, $X = 1$, $X = 2$, and $X = 3$.

   $P(X = 0) = \binom{17}{0} (0.2)^0 (0.8)^{17} = 0.0387$

   $P(X = 1) = \binom{17}{1} (0.2)^1 (0.8)^{16} = 0.1645$

   $P(X = 2) = \binom{17}{2} (0.2)^2 (0.8)^{15} = 0.3085$

   $P(X = 3) = \binom{17}{3} (0.2)^3 (0.8)^{14} = 0.2701$

2. Sum these probabilities:

   $P(X \leq 3) = P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3)$

   $P(X \leq 3) = 0.0387 + 0.1645 + 0.3085 + 0.2701 = 0.7818$

3. Find the complement:

   $P(X \geq 4) = 1 - P(X \leq 3) = 1 - 0.7818 = 0.2182$

Final Answer:

The probability that at least 4 flights arrive late is approximately 0.2182 (or 21.82%).

---

Do you want more details on any of the steps, or have any questions?

Here are 5 related questions:

1. What is the probability that exactly 5 flights arrive late?
2. How would the probability change if 10% of flights were late?
3. What is the expected number of late flights for Southwest Air?
4. How does the binomial distribution change if we increase the number of trials (flights)?
5. What are the odds of having all 17 flights arrive on time?

Tip: Using the complement rule is often simpler when calculating "at least" probabilities in binomial distributions!