This problem can be solved using the binomial distribution since each flight is either on time or not, with a fixed probability of being on time.

Given:

• Probability of a flight being on time ($p$) = 0.75
• Number of flights ($n$) = 20
• We are interested in the probability of the number of on-time flights ($X$) being between 9 and 11 inclusive.

We need to calculate $P(9 \leq X \leq 11)$, where $X$ follows a binomial distribution: $X \sim \text{Binomial}(n = 20, p = 0.75)$.

Step-by-Step Solution:

1. Define the Range: We want to find $P(9 \leq X \leq 11) = P(X = 9) + P(X = 10) + P(X = 11)$.

2. Use the Binomial Probability Formula: $P(X = k) = \binom{n}{k} p^k (1 - p)^{n - k}$ where $\binom{n}{k}$ is the binomial coefficient, $p$ is the probability of a successful outcome, and $(1 - p)$ is the probability of a failure.

3. Calculate Each Probability: We can compute $P(X = 9)$, $P(X = 10)$, and $P(X = 11)$ individually using the binomial formula.

   • $P(X = 9) = \binom{20}{9} (0.75)^9 (0.25)^{11}$
   • $P(X = 10) = \binom{20}{10} (0.75)^{10} (0.25)^{10}$
   • $P(X = 11) = \binom{20}{11} (0.75)^{11} (0.25)^9$

Calculations

Let me calculate these probabilities for you:from scipy.stats import binom

Given data

n = 20 # number of trials (flights) p = 0.75 # probability of success (flight on time)

Calculate probabilities for X = 9, X = 10, X = 11

p_9 = binom.pmf(9, n, p) p_10 = binom.pmf(10, n, p) p_11 = binom.pmf(11, n, p)

Total probability for 9 <= X <= 11

total_probability = p_9 + p_10 + p_11 total_probability0.039989776127185920.03998977612718592successThe probability that between 9 and 11 flights (inclusive) are on time out of 20 flights is approximately $0.03999$ or 3.999%.

Would you like more details on this calculation, or do you have any other questions?

Here are some related questions to consider:

1. How would the probability change if the percentage of on-time flights increased?
2. What is the probability of having exactly 15 flights on time in this scenario?
3. How would you calculate the expected number of on-time flights out of 20?
4. What if we wanted the probability of having more than 15 flights on time?
5. How does the variance of on-time flights relate to the probability in this scenario?

Tip: For problems involving a large number of trials, the normal approximation to the binomial distribution can simplify calculations.