**Your flight has been delayed:** At Denver International Airport, 

75%

 of recent flights have arrived on time. A sample of 

13

 flights is studied. Round the probabilities to at least four decimal places.

Part 1 of 4

(a)Find the probability that all 

13

 of the flights were on time.

The problem provided involves finding the probability that all 13 flights were on time, given that the probability of any single flight being on time is 75%.

We can model this as a binomial probability problem, where:

- The number of trials \(n = 13\) (the number of flights).
- The probability of success (a flight being on time) \(p = 0.75\).
- The probability of failure (a flight being delayed) \(q = 1 - p = 0.25\).

The probability that **all 13 flights were on time** can be calculated using the binomial distribution formula for exactly \( k = 13 \) successes:

\[
P(\text{all 13 on time}) = p^{13}
\]

Let me calculate this.

\[
P(\text{all 13 on time}) = 0.75^{13}
\]

Let me compute that for you.The probability that all 13 flights were on time is approximately **0.0238** or 2.38%.

Would you like further details on how this was calculated or explanations about any part of the process? Here are five related questions that might interest you:

1. What is the probability that exactly 12 flights were on time?
2. How would you calculate the probability that fewer than 10 flights were on time?
3. What is the expected number of on-time flights out of 13?
4. How do binomial distributions apply to real-world scenarios like this?
5. What would the probability be if the sample size increased to 20 flights?

**Tip:** In binomial probability, as the number of trials increases, the normal approximation to the binomial distribution becomes more accurate.

Your flight has been delayed: At Denver International Airport, 75% of recent flights have arrived on time. A sample of 13 flights is studied. Find the probability that all 13 of the flights were on time.

Calculate the probability that all 13 flights arrive on time given a 75% on-time probability. Explore binomial probability concepts and apply the formula for success probability to real-world situations. Suitable for Grades 10-12 students.

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