From experience, an airline knows that only 70% of the passengers booked for a certain flight actually show up. If  passengers are randomly selected, find the probability that at most 5 of them show up.
Carry your intermediate computations to at least four decimal places, and round your answer to two decimal places.

We can model this situation using the **binomial distribution**. The number of passengers showing up follows a binomial distribution because each passenger either shows up or does not (two possible outcomes), and each passenger is independent of the others.

The binomial distribution is defined as:

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

where:
- \( n \) is the number of passengers booked.
- \( p \) is the probability that a passenger shows up (in this case, 0.70).
- \( k \) is the number of passengers who show up.
- \( \binom{n}{k} \) is the binomial coefficient: \( \frac{n!}{k!(n-k)!} \).

We are asked to find the probability that at most 5 passengers show up, which means calculating:

\[
P(X \leq 5) = P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3) + P(X = 4) + P(X = 5)
\]

### Steps:

1. **Identify the parameters:**
   - The probability of a passenger showing up: \( p = 0.70 \)
   - The probability of a passenger not showing up: \( 1 - p = 0.30 \)
   - The number of trials (passengers booked): \( n \) (but this value is not provided directly, so if it's missing we cannot continue).

Could you please specify the number of passengers booked? That way, I can proceed with the calculations.

From experience, an airline knows that only 70% of the passengers booked for a certain flight actually show up. If passengers are randomly selected, find the probability that at most 5 of them show up. Carry your intermediate computations to at least four decimal places, and round your answer to two decimal places.

Learn how to calculate the probability that at most 5 passengers show up for a flight when 70% of booked passengers typically appear. This problem utilizes the binomial distribution formula and covers key probability concepts. Step-by-step guidance provided, suitable for Grades 11-12 and college students.

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