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.