Let's go through each part of the problem shown in the image to address the errors and verify correct values.

Problem Setup:

We are dealing with a Binomial Distribution, $X \sim B(n=24, p=0.17)$, where:

• $n = 24$ (the number of patients),
• $p = 0.17$ (the probability a patient actually has the flu).

1. Probability that exactly 9 patients have the flu:

The formula for a binomial probability is:

$P(X = k) = \binom{n}{k} \cdot p^k \cdot (1-p)^{n-k}$

Where:

• $n = 24$,
• $k = 9$,
• $p = 0.17$.

Let me calculate that.The correct probability that exactly 9 out of 24 patients actually have the flu is approximately 0.00948, not 0.08776 as shown in the image.

2. Probability that fewer than 3 of the 24 patients actually have the flu:

This involves calculating the sum of the probabilities for $X = 0$, $X = 1$, and $X = 2$:

$P(X < 3) = P(X = 0) + P(X = 1) + P(X = 2)$

Let me calculate this.The correct probability that fewer than 3 out of 24 patients actually have the flu is approximately 0.19988, not 0.2625 as shown.

3. Mean of the Distribution:

The mean of a binomial distribution is given by:

$\mu = n \cdot p$

Substituting the given values $n = 24$ and $p = 0.17$:

$\mu = 24 \cdot 0.17 = 4.08$

This is correct as shown in the image.

4. Standard Deviation of the Distribution:

The standard deviation of a binomial distribution is:

$\sigma = \sqrt{n \cdot p \cdot (1 - p)}$

Let me calculate that for you.The correct standard deviation of the distribution is approximately 1.84, not 1.83 as shown in the image.

Summary of corrections:

1. Probability that exactly 9 patients have the flu: 0.00948 (corrected from 0.08776).
2. Probability that fewer than 3 patients have the flu: 0.19988 (corrected from 0.2625).
3. Mean of the distribution: 4.08 (correct).
4. Standard deviation of the distribution: 1.84 (corrected from 1.83).

Would you like to go over any details or ask further questions?

Follow-up questions:

1. What is the probability that exactly 0 patients have the flu?
2. What is the cumulative probability that at most 5 patients have the flu?
3. How would the probability change if the flu probability increased to 25%?
4. Can you calculate the probability that exactly 12 patients have the flu?
5. What is the variance of this binomial distribution?

Tip: For binomial distributions, always check if using a normal approximation makes sense when $n$ is large and $p$ is not too close to 0 or 1.