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Part 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 eleven​,

and​ (c) less than twelve.

This is a problem about **binomial probability**, as it involves a fixed number of independent trials (16 consumers), each of which results in one of two outcomes (either the consumer says it's important or they don't). 

The key components for binomial probability are:
- \( n \): number of trials (in this case, 16 consumers),
- \( p \): probability of success (here, 45% or 0.45),
- \( q = 1 - p \): probability of failure (here, \( 1 - 0.45 = 0.55 \)),
- \( X \): the number of successes (consumers who say it's important).

The binomial probability formula is:

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

Where:
- \( \binom{n}{k} \) is the binomial coefficient, calculated as \( \frac{n!}{k!(n-k)!} \),
- \( p^k \) is the probability of exactly \( k \) successes,
- \( q^{n-k} \) is the probability of the remaining trials being failures.

Let's solve the problem step by step.

### Part (a) - Find the probability that exactly 7 consumers say it is important
We need to calculate \( P(X = 7) \).

Using \( n = 16 \), \( p = 0.45 \), and \( q = 0.55 \):

\[
P(X = 7) = \binom{16}{7} \cdot (0.45)^7 \cdot (0.55)^9
\]

### Part (b) - Find the probability that at least 11 consumers say it is important
We need to calculate \( P(X \geq 11) \), which is the sum of the probabilities from 11 to 16.

\[
P(X \geq 11) = P(X = 11) + P(X = 12) + P(X = 13) + P(X = 14) + P(X = 15) + P(X = 16)
\]

### Part (c) - Find the probability that fewer than 12 consumers say it is important
We need to calculate \( P(X < 12) \), which is the sum of the probabilities from 0 to 11.

\[
P(X < 12) = P(X = 0) + P(X = 1) + \cdots + P(X = 11)
\]

I will now calculate these values for you.Here are the results for the given problem:

### (a) The probability that exactly 7 consumers say it is important is:
\[
P(X = 7) \approx 0.1969
\]

### (b) The probability that at least 11 consumers say it is important is:
\[
P(X \geq 11) \approx 0.0486
\]

### (c) The probability that fewer than 12 consumers say it is important is:
\[
P(X < 12) \approx 0.9851
\]

Would you like further details or have any questions? Here are some related questions to explore further:

1. How does the probability change if the number of consumers increases to 20?
2. What happens if the probability that a consumer says it's important changes to 50%?
3. How is the binomial distribution connected to the normal distribution?
4. What is the mean and variance of this binomial distribution?
5. How would the probabilities change if you were sampling with a different probability (e.g., 30%)?

**Tip:** The sum of all probabilities in a binomial distribution always equals 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 eleven, and (c) less than twelve.

Calculate the binomial probability for consumer choices about ethical clothing. This problem involves key probability concepts, such as binomial distribution, and uses a real-world scenario to explore probabilities. Suitable for Grades 10-12.

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