https://file.aiquickdraw.com/mathbot/file/17294687595716b5nl7u5.jpg

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?

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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.

American Airlines' flights from Denver to Dallas are on time 80% of the time. Suppose 15 flights are randomly selected, and the number of on-time flights is recorded. What is the probability that at least 11 flights, at most 7 flights, and exactly 9 flights are on time?

Explore the binomial distribution and probability of at least 11, at most 7, and exactly 9 out of 15 flights being on time. Learn about the formula P(X = k) and how it applies to this real-world problem.

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