The image provided is a normal distribution graph with mean $\mu = 0$ and standard deviation $\sigma = 1$. The shaded area between $x = -1$ and $x = 1$ represents the probability $P(-1 \leq X \leq 1) = 0.6827$, corresponding to about 68.27% of the data within one standard deviation of the mean in a standard normal distribution.

Below this graph, the problem involves a binomial distribution for flights from Denver to Dallas, where 80% of the flights are on time. The sample size is 15 flights, and three questions are posed regarding probabilities based on the number of on-time flights.

Let's break down each question:

1. The probability that at least 11 flights are on time: This can be modeled using the binomial distribution with:

   • $n = 15$ (number of trials or flights),
   • $p = 0.80$ (probability of success, i.e., flights being on time).

   We need to find $P(X \geq 11)$, where $X$ is the number of on-time flights. This is the sum of probabilities $P(X = 11) + P(X = 12) + P(X = 13) + P(X = 14) + P(X = 15)$.

2. The probability that at most 7 flights are on time: Here, you are asked to find $P(X \leq 7)$, which is the cumulative probability for having 7 or fewer on-time flights.

3. The probability that exactly 9 flights are on time: This is simply $P(X = 9)$ from the binomial distribution.

These probabilities can be computed using the binomial formula: $P(X = k) = \binom{n}{k} p^k (1-p)^{n-k}$ or using a statistical calculator or software for cumulative probabilities.

Would you like me to calculate these probabilities for you, or provide step-by-step guidance on solving them?

In the meantime, here are 5 related questions to extend your understanding:

1. What is the expected number of on-time flights out of 15 flights?
2. How do you calculate the variance and standard deviation in a binomial distribution?
3. What is the probability that fewer than 5 flights are on time?
4. How would the probabilities change if the on-time rate dropped to 70%?
5. What is the difference between a binomial distribution and a normal distribution?

Tip: For binomial probabilities, it's helpful to use a cumulative distribution function (CDF) table or a binomial calculator for faster results when $n$ is large.