It looks like you've uploaded a file related to a probability question. Based on the context, it seems to be a problem involving the binomial distribution.

The problem asks you to find the probability that a certain number of consumers believe that clothing should be made without child labor, with the given conditions:

• The probability that a consumer thinks it's important is 0.45.
• You randomly select 18 consumers.
• The question requires you to calculate the probability for different scenarios, including exactly 8 consumers, at least 10, and fewer than 12.

This is a classic binomial probability problem. In general, for a binomial distribution:

• n = number of trials (18 consumers)
• p = probability of success (0.45)
• k = number of successes (number of consumers who think it’s important)

For part (a), you need to calculate $P(X = 8)$, which can be done using 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 is computed as:

$\binom{n}{k} = \frac{n!}{k!(n-k)!}$

Let me calculate this probability for $P(X = 8)$. math

Given values

n = 18 # number of trials p = 0.45 # probability of success k = 8 # number of successes

Binomial coefficient

binomial_coeff = math.comb(n, k)

Binomial probability formula

probability_exactly_8 = binomial_coeff * (p ** k) * ((1 - p) ** (n - k))

Round the result to three decimal places

probability_exactly_8_rounded = round(probability_exactly_8, 3)

probability_exactly_8_roundedThe probability that exactly 8 consumers say it is important that the clothing they buy is made without child labor is approximately $P(\text{exactly 8}) = 0.186$.

Would you like to proceed with parts (b) and (c) of the problem, or do you have any questions about this result?

Here are some related questions for you:

1. How would you calculate the probability for exactly 10 consumers?
2. How would the formula change if the number of trials (n) were different?
3. What is the difference between a binomial distribution and a normal distribution?
4. How do you interpret the binomial coefficient in this context?
5. What would happen to the probabilities if the probability of success (p) were higher than 0.45?

Tip: Remember to apply the binomial probability formula correctly for each specific value of $k$.