​Forty-five percent of consumers say it is important that the clothing they buy is made without child labor. You randomly select 16 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 seven​, ​(b) at least nine​, and​ (c) less than ten.
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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 seven is Upper P left parenthesis exactly seven right parenthesisequals
  
enter your response here.
​(Round to three decimal places as​ needed.)

To solve this problem, we'll use the **binomial probability formula**. The situation described involves a fixed number of trials (16 consumers), with a probability of success (saying it's important that the clothing is made without child labor) equal to 0.45. The probability of a specific number of successes follows a binomial distribution.

### Binomial Probability Formula
The formula to find the probability of exactly \( k \) successes in \( n \) trials is:
\[
P(X = k) = \binom{n}{k} p^k (1-p)^{n-k}
\]
where:
- \( \binom{n}{k} = \frac{n!}{k!(n-k)!} \) is the binomial coefficient.
- \( n \) is the number of trials.
- \( k \) is the number of successes.
- \( p \) is the probability of success.
- \( (1-p) \) is the probability of failure.

Given:
- \( n = 16 \)
- \( p = 0.45 \)
- \( k = 7 \)

### Part (a): Probability of exactly 7 successes

Let's calculate the probability of exactly 7 consumers saying it's important.

\[
P(X = 7) = \binom{16}{7} (0.45)^7 (1 - 0.45)^{16-7}
\]
Breaking it down:
- \( \binom{16}{7} = \frac{16!}{7!(16-7)!} = \frac{16!}{7! \cdot 9!} \)
- \( (0.45)^7 \) is the probability of 7 successes.
- \( (0.55)^9 \) is the probability of 9 failures.

I will calculate these values to give you the exact probability rounded to three decimal places.The probability that exactly 7 consumers say it is important that the clothing they buy is made without child labor is approximately **0.197**.

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

Here are some related questions to further explore the topic:

1. How do you calculate the probability of at least 9 consumers using the binomial distribution?
2. What is the cumulative probability formula for calculating probabilities less than or equal to a certain number?
3. How does changing the sample size (number of consumers) affect the probability of a specific outcome?
4. What if the probability of a consumer caring about child labor changes to 0.60? How would that influence our results?
5. How would you interpret the result of this probability in a real-world context?

**Tip:** In binomial probability problems, it's always good to double-check your \( p \) and \( q \) values since they must add up to 1.

Forty-five percent of consumers say it is important that the clothing they buy is made without child labor. You randomly select 16 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 seven​, ​(b) at least nine​, and​ (c) less than ten.

This problem explores the use of binomial probability to find the likelihood that a certain number of consumers believe ethical clothing is important. Using a 45% success rate and a sample size of 16, the solution walks through calculating probabilities for exactly 7, at least 9, and fewer than 10 consumers. This is ideal for students in Grades 11-12 studying probability theory.

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