​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 six​, ​(b) at least ten​, and​ (c) less than eleven.
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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 six is Upper P left parenthesis exactly six right parenthesisequals
  
enter your response here.
​(Round to three decimal places as​ needed.)
Part 2
​(b) The probability that the number of consumers who say it is important that the clothing they buy is made without child labor is at least ten is Upper P left parenthesis at least ten right parenthesisequals
  
enter your response here.
​(Round to three decimal places as​ needed.)
Part 3
​(c) The probability that the number of consumers who say it is important that the clothing they buy is made without child labor is less than eleven is Upper P left parenthesis less than eleven right parenthesisequals
  
enter your response here.
​(Round to three decimal places as​ needed.)

To solve this problem, we can model it using the **binomial distribution**. The binomial distribution is appropriate here because we have a fixed number of independent trials (16 consumers), each with two possible outcomes (whether they say it is important or not), and a constant probability of success (45%, or 0.45).

Given:
- \( n = 16 \) (number of trials)
- \( p = 0.45 \) (probability of success)

The binomial probability formula is:

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

where:
- \( P(X = k) \) is the probability of exactly \( k \) successes,
- \( \binom{n}{k} \) is the binomial coefficient, calculated as \( \frac{n!}{k!(n-k)!} \),
- \( p \) is the probability of success on each trial, and
- \( (1-p) \) is the probability of failure on each trial.

### Part 1: Probability of exactly six successes
We need to find \( P(X = 6) \):

\[
P(X = 6) = \binom{16}{6} (0.45)^6 (0.55)^{10}
\]

### Part 2: Probability of at least ten successes
We need to find \( P(X \geq 10) \), which is the sum of the probabilities from 10 to 16:

\[
P(X \geq 10) = P(X = 10) + P(X = 11) + \dots + P(X = 16)
\]

### Part 3: Probability of less than eleven successes
We need to find \( P(X < 11) \), which is the sum of the probabilities from 0 to 10:

\[
P(X < 11) = P(X = 0) + P(X = 1) + \dots + P(X = 10)
\]

Let's calculate these probabilities.Here are the probabilities rounded to three decimal places:

- **(a)** The probability that exactly 6 consumers say it is important that the clothing they buy is made without child labor is \( \mathbf{0.168} \).
- **(b)** The probability that at least 10 consumers say it is important is \( \mathbf{0.124} \).
- **(c)** The probability that fewer than 11 consumers say it is important is \( \mathbf{0.951} \).

Would you like any further details or have any questions?

Here are some related questions you might be interested in:
1. What is the probability that exactly 8 consumers say it is important?
2. How would the probabilities change if the sample size were increased to 20 consumers?
3. What is the expected number of consumers who say it is important that the clothing is made without child labor?
4. How does the variance of the distribution change if the probability of success increases to 0.6?
5. What is the probability that fewer than 5 consumers say it is important?

**Tip:** For large sample sizes, the binomial distribution can be approximated by a normal distribution.

Explore the probability that consumers prefer clothing made without child labor based on a random sample. Calculate exact probabilities and explore scenarios using the binomial distribution. Detailed solutions for exactly six, at least ten, and less than eleven consumers.

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