To solve this problem, we need to use the binomial probability formula since we're dealing with a fixed number of independent trials, where each trial has two possible outcomes: either a passenger shows up or they do not.

The binomial probability formula is given by:

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

where:

• $n$ is the number of trials (in this case, 6 passengers),
• $k$ is the number of successes (the number of passengers who show up),
• $p$ is the probability of success on an individual trial (0.85, since 85% of the passengers show up),
• $\binom{n}{k}$ is the binomial coefficient, calculated as $\frac{n!}{k!(n-k)!}$.

We want to find the probability that at least 4 of the 6 passengers show up. This means we need to calculate the sum of the probabilities for 4, 5, and 6 passengers showing up.

$P(\text{at least 4 show up}) = P(X = 4) + P(X = 5) + P(X = 6)$

Step 1: Calculate $P(X = 4)$

$P(X = 4) = \binom{6}{4} \cdot (0.85)^4 \cdot (0.15)^2$

Step 2: Calculate $P(X = 5)$

$P(X = 5) = \binom{6}{5} \cdot (0.85)^5 \cdot (0.15)^1$

Step 3: Calculate $P(X = 6)$

$P(X = 6) = \binom{6}{6} \cdot (0.85)^6 \cdot (0.15)^0$

Step 4: Sum the probabilities

$P(\text{at least 4 show up}) = P(X = 4) + P(X = 5) + P(X = 6)$

Let me calculate these values.The probability that at least 4 out of the 6 passengers show up is approximately 0.9527, or 95.27%.

Would you like more details on the calculations or have any other questions?

Here are some related questions:

1. How is the binomial coefficient calculated in combinatorics?
2. What is the significance of the binomial distribution in probability theory?
3. How would the probability change if the number of passengers selected was increased?
4. What if the probability of a passenger showing up was only 70%? How would that affect the results?
5. Can you explain how to derive the binomial probability formula?
6. How does the concept of "at least" affect the calculation of probabilities?
7. What are some real-world applications of the binomial distribution?
8. How does the law of large numbers relate to binomial probabilities?

Tip: When dealing with probabilities involving "at least" or "at most," it's often easier to use the complement rule, especially when the range of successful outcomes is broad.