Data collected by the Substance Abuse and Mental Health Services Administration (SAMSHA) suggests that 69.7% of 18-20 year olds consumed alcoholic beverages in 2008.Calculate the probability that exactly 6 out of 10 randomly sampled 18- 20 year olds consumed an alcoholic drink.

To calculate the probability that exactly 6 out of 10 randomly sampled 18-20 year olds consumed an alcoholic beverage, we use the **binomial probability formula**. The binomial distribution is appropriate here because we are dealing with two outcomes (either they consumed alcohol or they did not).

The binomial probability formula is:

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

Where:
- \(n\) = the number of trials (in this case, 10 people)
- \(k\) = the number of successes (in this case, 6 people)
- \(p\) = the probability of success on a single trial (here, 69.7% or 0.697)
- \(\binom{n}{k}\) = "n choose k", or the number of ways to choose \(k\) successes out of \(n\) trials

### Step-by-step calculation:

1. **Identify the variables:**
   - \(n = 10\)
   - \(k = 6\)
   - \(p = 0.697\)

2. **Calculate the binomial coefficient:**
   \[
   \binom{10}{6} = \frac{10!}{6!(10-6)!} = \frac{10!}{6!4!} = \frac{10 \times 9 \times 8 \times 7}{4 \times 3 \times 2 \times 1} = 210
   \]

3. **Compute the probability:**
   \[
   P(X = 6) = 210 \times (0.697)^6 \times (1 - 0.697)^4
   \]
   \[
   = 210 \times (0.697)^6 \times (0.303)^4
   \]

   Let's break it down:
   \[
   (0.697)^6 \approx 0.0845
   \]
   \[
   (0.303)^4 \approx 0.0084
   \]

4. **Multiply all values:**
   \[
   P(X = 6) = 210 \times 0.0845 \times 0.0084 \approx 210 \times 0.0007113 = 0.1494
   \]

So, the probability that exactly 6 out of 10 randomly sampled 18-20 year olds consumed an alcoholic beverage is approximately **0.1494**, or about **14.94%**.

---

Do you want further details or explanations on any of the steps? Here are 5 related questions to think about:

1. How would the probability change if the sample size increased to 20?
2. What is the probability that fewer than 5 out of 10 consumed alcohol?
3. How does the binomial distribution apply in other real-life scenarios?
4. What is the probability that all 10 sampled individuals consumed alcohol?
5. How would the results differ if the probability of consumption were 50% instead of 69.7%?

**Tip:** The sum of probabilities for all possible outcomes (from 0 to 10 successes) in a binomial distribution always equals 1!

Data collected by the Substance Abuse and Mental Health Services Administration (SAMSHA) suggests that 69.7% of 18-20 year olds consumed alcoholic beverages in 2008. Calculate the probability that exactly 6 out of 10 randomly sampled 18-20 year olds consumed an alcoholic drink.

Calculate the probability that exactly 6 out of 10 randomly sampled 18-20 year olds consumed alcohol using the binomial distribution formula. This problem covers key probability concepts such as binomial coefficients and the binomial theorem. The solution is broken down step-by-step and is suitable for students in Grades 10-12.

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