Data collected by the National Center for Health Statistics showed that 23.53% of adults were told at some point in their life they have hypertension. If we select a simple random sample of 20 U.S. adults and assume that the probability that each has been told they have hypertension is .24. 

	Find the probability that the number of people in the sample who have been told that they have hypertension will be 3 (exactly)


This problem involves finding the probability of a specific number of people in a sample being told they have hypertension. Given that we are selecting a random sample of 20 U.S. adults and assuming the probability that each person has been told they have hypertension is 0.24, we can use the **binomial probability formula**:

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

Where:
- \( n = 20 \) (sample size),
- \( k = 3 \) (the number of people in the sample who were told they have hypertension),
- \( p = 0.24 \) (the probability that a person has been told they have hypertension),
- \( \binom{n}{k} \) is the binomial coefficient, which is calculated as \( \frac{n!}{k!(n-k)!} \).

Let's calculate this probability step by step:

1. **Calculate the binomial coefficient** \(\binom{n}{k}\):
   \[
   \binom{20}{3} = \frac{20!}{3!(20 - 3)!} = \frac{20 \times 19 \times 18}{3 \times 2 \times 1} = 1140
   \]

2. **Calculate the probability terms**:
   \[
   p^k = (0.24)^3 = 0.24 \times 0.24 \times 0.24 = 0.013824
   \]
   \[
   (1 - p)^{n - k} = (0.76)^{17} = 0.0526061 \quad (\text{approximately})
   \]

3. **Now, multiply everything**:
   \[
   P(X = 3) = 1140 \times 0.013824 \times 0.0526061 \approx 0.8283
   \]

So, the probability that exactly 3 people in the sample have been told they have hypertension is approximately **0.08283**, or about **8.28%**.

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Would you like any more details or have questions about this?

Here are 5 related questions for further exploration:
1. What is the probability that exactly 5 people in the sample have been told they have hypertension?
2. How would the probability change if the sample size were 25 instead of 20?
3. What is the probability that no one in the sample has been told they have hypertension?
4. How would we calculate the probability that fewer than 3 people in the sample have been told they have hypertension?
5. What is the expected number of people in the sample who have been told they have hypertension?

**Tip**: When dealing with binomial distributions, you can use cumulative binomial probabilities for questions involving "at least" or "fewer than" scenarios.

Data collected by the National Center for Health Statistics showed that 23.53% of adults were told at some point in their life they have hypertension. If we select a simple random sample of 20 U.S. adults and assume that the probability that each has been told they have hypertension is 0.24, find the probability that the number of people in the sample who have been told they have hypertension will be 3 (exactly).

Learn how to calculate the binomial probability of exactly 3 adults out of a sample of 20 being told they have hypertension. Using the binomial distribution formula, this problem involves a probability of 0.24 and explores key concepts in probability. Suitable for students in Grades 10-12.

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