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.