https://file.aiquickdraw.com/mathbot/file/1726980748559ob3com18.jpg

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.

A nurse has found that when a patient calls the medical advice line claiming to have the flu, there is a 17% chance that the patient truly has the flu. Of the next 24 patients, we are interested in how many actually have the flu.

Solve a binomial distribution problem where 24 patients call a medical line claiming to have the flu, with a 17% chance that they truly have the flu. Calculate the probability of exactly 9 patients having the flu, fewer than 3, and determine the mean and standard deviation of the distribution.

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