​Forty-five percent of consumers say it is important that the clothing they buy is made without child labor. You randomly select 17 consumers. Find the probability that the number of consumers who say it is important that the clothing they buy is made without child labor is​ (a) exactly five​, ​(b) at least ten​, and​ (c) less than twelve.
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Part 1
​(a) The probability that the number of consumers who say it is important that the clothing they buy is made without child labor is exactly five is Upper P left parenthesis exactly five right parenthesisequals
  
enter your response here.
​(Round to three decimal places as​ needed.)

This problem is modeled as a **binomial distribution** since we are dealing with a fixed number of independent trials (selecting 17 consumers) and two possible outcomes for each trial (either a consumer says it is important or not).

### Key parameters:
- \( n = 17 \) (the number of consumers)
- \( p = 0.45 \) (the probability that a consumer says it is important)
- \( q = 1 - p = 0.55 \) (the probability that a consumer does not say it is important)

The probability of getting exactly \( k \) successes (in this case, exactly 5 consumers saying it's important) in a binomial distribution is given by the **binomial probability formula**:

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

where \( \binom{n}{k} \) is the binomial coefficient, which represents the number of ways to choose \( k \) successes out of \( n \) trials.

Now, let's solve part (a).

### (a) Probability that exactly 5 consumers say it's important

We use the binomial formula with \( k = 5 \), \( n = 17 \), and \( p = 0.45 \):

\[
P(X = 5) = \binom{17}{5} (0.45)^5 (0.55)^{12}
\]

Let me calculate this.The probability that exactly 5 consumers say it is important that the clothing they buy is made without child labor is approximately \( P(X = 5) = 0.087 \).

Now, would you like to proceed with parts (b) and (c), or need further details on the current calculation?

Here are some related questions:
1. What is the probability of selecting fewer than 5 consumers who say it is important?
2. What is the expected number of consumers who say it is important?
3. How does the probability change if the number of consumers increases to 20?
4. What happens to the probability if the percentage of people caring about child labor changes to 50%?
5. How would you calculate the cumulative probability for more than 5 consumers?

**Tip:** When calculating binomial probabilities, the cumulative distribution function (CDF) can be helpful for "at least" or "less than" types of questions.

Forty-five percent of consumers say it is important that the clothing they buy is made without child labor. You randomly select 17 consumers. Find the probability that the number of consumers who say it is important that the clothing they buy is made without child labor is (a) exactly five, (b) at least ten, and (c) less than twelve.

Learn how to solve a binomial probability problem where 45% of consumers say it is important that the clothing they buy is made without child labor. This solution demonstrates how to calculate the probability of exactly 5, at least 10, and fewer than 12 consumers in a sample of 17. Detailed steps include the binomial formula and its application. Suitable for Grades 11-12.

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