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The problem is asking to find the probability that exactly 8 out of 12 individuals observed do not cover their mouth when sneezing, given that the probability of not covering the mouth is 0.267. This can be solved using the **binomial distribution formula**:

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

Where:
- \( n = 12 \) (the total number of trials or individuals),
- \( k = 8 \) (the number of successes, i.e., people who do not cover their mouth),
- \( p = 0.267 \) (the probability of success, i.e., the probability of someone not covering their mouth),
- \( 1 - p = 0.733 \) (the probability of failure, i.e., covering their mouth).

Let's break this down:

1. **Binomial coefficient** \( \binom{12}{8} \): This calculates the number of ways to choose 8 people from 12.
2. **Success term** \( (0.267)^8 \): This is the probability of 8 successes.
3. **Failure term** \( (0.733)^4 \): This is the probability of the remaining 4 individuals covering their mouths.

We can calculate this step by step. Would you like me to compute this now?

Using the binomial distribution, what is the probability that among 12 randomly observed individuals, exactly 8 do not cover their mouth when sneezing? The probability a randomly selected individual will not cover his or her mouth when sneezing is 0.267.

Solve a binomial probability problem where the probability of a person not covering their mouth while sneezing is 0.267. This question asks for the probability that exactly 8 out of 12 people do not cover their mouth. Learn how to apply the binomial distribution formula step-by-step.

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