For the US population about 20% get the flu.  I had a class where 8 people told me that they got the flu.  What is the probability that if I pick 32 students (the size of this class) I get 8 or more that got the flu?

This problem is asking about the probability of getting 8 or more students who contracted the flu in a class of 32, given that about 20% of the population gets the flu. To solve this, we can use the **binomial distribution**, since we're dealing with a fixed number of trials (32 students) and two possible outcomes for each trial (got the flu or did not get the flu).

### Step 1: Define the problem in terms of binomial distribution
The binomial distribution is appropriate here because:
- Each student either got the flu or didn’t (binary outcomes).
- The probability \( p \) of a student getting the flu is 0.2 (20%).
- The number of trials \( n \) is 32 students.
- We're interested in the probability of getting 8 or more students who got the flu.

The binomial probability mass function (PMF) is given by:

\[
P(X = k) = \binom{n}{k} p^k (1 - p)^{n - k}
\]

where:
- \( n = 32 \) (the number of trials),
- \( p = 0.2 \) (the probability of a student getting the flu),
- \( k \) is the number of students who got the flu.

### Step 2: Set up the probability calculation
We need to find the probability that 8 or more students got the flu, i.e., \( P(X \geq 8) \).

This is equivalent to summing up the probabilities for \( X = 8 \) to \( X = 32 \), or we can subtract the probability of \( X \leq 7 \) from 1:

\[
P(X \geq 8) = 1 - P(X \leq 7)
\]

### Step 3: Calculate the probability
We can now calculate the cumulative probability \( P(X \leq 7) \) and subtract that from 1.

Let me calculate that for you.The probability of getting 8 or more students who contracted the flu in a class of 32 students is approximately **0.302** or **30.2%**.

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

Here are 5 related questions:
1. What is the probability of getting exactly 8 students with the flu?
2. How does the probability change if the flu rate is 25% instead of 20%?
3. What if the class size increases to 40 students? How does that affect the probability?
4. Can the binomial distribution be approximated using the normal distribution in this case?
5. What is the expected number of students in a class of 32 to get the flu?

**Tip:** For large sample sizes and probabilities not too close to 0 or 1, the binomial distribution can be approximated using the normal distribution to simplify calculations.

For the US population about 20% get the flu. I had a class where 8 people told me that they got the flu. What is the probability that if I pick 32 students (the size of this class) I get 8 or more that got the flu?

Find out the probability that 8 or more students in a class of 32 have the flu, given a 20% flu rate in the US population. This problem uses the binomial distribution and covers key concepts in probability. Suitable for students in Grades 11-12.

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