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This problem involves determining the probability that at least 6 of the 8 passengers show up, given that the probability of any one passenger showing up is 80%. We can solve this using the **binomial distribution** formula.

### Step-by-Step Solution

1. **Identify the key parameters**:
   - \( n = 8 \): Number of passengers.
   - \( p = 0.8 \): Probability that each passenger shows up.
   - \( X \): Random variable representing the number of passengers that show up.
   
   We need to find \( P(X \geq 6) \).

2. **Express \( P(X \geq 6) \)**:
   \[
   P(X \geq 6) = P(X = 6) + P(X = 7) + P(X = 8)
   \]
   This is because we need the probability of 6, 7, or 8 passengers showing up.

3. **Use the Binomial Probability Formula**:
   The binomial probability formula is:
   \[
   P(X = k) = \binom{n}{k} p^k (1 - p)^{n - k}
   \]
   where \(\binom{n}{k}\) represents the combination, i.e. the number of ways to choose \( k \) successes out of \( n \) trials, calculated as:
   \[
   \binom{n}{k} = \frac{n!}{k!(n-k)!}
   \]

4. **Calculate each required probability**:

   - For \( P(X = 6) \):
     \[
     P(X = 6) = \binom{8}{6} (0.8)^6 (0.2)^2
     \]
     \[
     = \frac{8!}{6!2!} (0.8)^6 (0.2)^2 = 28 \times (0.8)^6 \times (0.2)^2
     \]
     \[
     = 28 \times 0.262144 \times 0.04 = 0.2936
     \]

   - For \( P(X = 7) \):
     \[
     P(X = 7) = \binom{8}{7} (0.8)^7 (0.2)^1
     \]
     \[
     = 8 \times (0.8)^7 \times (0.2) = 8 \times 0.2097152 \times 0.2 = 0.3355
     \]

   - For \( P(X = 8) \):
     \[
     P(X = 8) = \binom{8}{8} (0.8)^8 (0.2)^0
     \]
     \[
     = 1 \times (0.8)^8 = 0.1678
     \]

5. **Add the probabilities**:
   \[
   P(X \geq 6) = P(X = 6) + P(X = 7) + P(X = 8)
   \]
   \[
   = 0.2936 + 0.3355 + 0.1678 = 0.7969
   \]

6. **Round to two decimal places**:
   \[
   P(X \geq 6) \approx 0.80
   \]

### Final Answer
The probability that at least 6 of the 8 passengers show up is **0.80**.

Would you like more details or have any questions?

Here are 5 related questions for further exploration:
1. What is the probability that exactly 5 passengers show up?
2. How would the probability change if the number of passengers was increased to 10?
3. How can we use cumulative distribution tables to find \( P(X \geq 6) \)?
4. What is the expected number of passengers that show up if 8 are booked?
5. How does the variance of a binomial distribution change as \( n \) increases?

**Tip:** When dealing with binomial probabilities, breaking down the problem into smaller calculations can make the solution more manageable.

From experience, an airline knows that only 80% of the passengers booked for a certain flight actually show up. If 8 passengers are randomly selected, find the probability that at least 6 of them show up.

Learn how to calculate the probability that at least 6 out of 8 passengers show up, using binomial distribution. This step-by-step guide walks through key concepts and formulas, suitable for Grades 10-12.

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