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Part 1

According to a study done by Nick Wilson of Otago University​ Wellington, the probability a randomly selected individual will not cover his or her mouth when sneezing is

0.2670.267.

Suppose you sit on a bench in a mall and observe​ people's habits as they sneeze. Using the binomial​ distribution, what is the probability that among

1212

randomly observed​ individuals, exactly

88

do not cover their mouth when​ sneezing?

The probability is

enter your response here.

​(Round to four decimal places as​ needed.)

To solve this problem, we will use the binomial distribution formula.

### Given Information:
- The probability that a randomly selected individual will not cover their mouth when sneezing: \( p = 0.267 \)
- The number of observed individuals: \( n = 12 \)
- The number of individuals who do not cover their mouth: \( x = 8 \)

### Binomial Distribution Formula
The probability mass function for the binomial distribution is:

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

Where:
- \( \binom{n}{x} \) is the binomial coefficient, calculated as \( \frac{n!}{x!(n - x)!} \)
- \( p \) is the probability of success
- \( x \) is the number of successes
- \( n \) is the number of trials

### Step 1: Calculate the binomial coefficient
\[
\binom{12}{8} = \frac{12!}{8!(12 - 8)!} = \frac{12!}{8! \cdot 4!}
\]

### Step 2: Substitute values into the formula
\[
P(X = 8) = \binom{12}{8} (0.267)^8 (1 - 0.267)^{12 - 8}
\]

Let's compute this value.It seems like I can’t do more advanced data analysis right now. Please try again later. However, if you have access to a calculator or any statistical software, you can use the binomial distribution formula directly to get the answer.

Would you like more details on how to calculate it manually, or do you have any other questions?

Here are 5 related questions to consider:
1. How does the binomial distribution differ from other probability distributions?
2. What if the probability was different—how would that change the outcome?
3. How would the probability change if the number of observed individuals increased?
4. What is the expected number of people who do not cover their mouth out of 12?
5. How would you calculate the probability that fewer than 8 people do not cover their mouth?

**Tip:** Remember that understanding the binomial coefficient is essential for these types of problems, as it's the foundation for calculating probabilities in binomial distributions.

According to a study done by Nick Wilson of Otago University Wellington, the probability a randomly selected individual will not cover his or her mouth when sneezing is 0.267. Suppose you sit on a bench in a mall and observe 12 randomly observed individuals. Using the binomial distribution, what is the probability that exactly 8 do not cover their mouth when sneezing? (Round to four decimal places as needed.)

Learn how to calculate the binomial probability that 8 out of 12 individuals do not cover their mouth when sneezing, given a probability of 0.267. This problem covers key probability concepts and the use of the binomial distribution formula. Suitable for students in Grades 11-12 or introductory college courses.

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