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. Complete parts​ (a) through​ (c).

Question content area bottom

Part 1

​(​a)

 Using the binomial​ distribution, what is the probability that among 

1010 

randomly observed​ individuals, exactly 

66 

do not cover their mouth when​ sneezing?

The probability is 

enter your response here.

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

The problem asks for the probability that exactly 6 out of 10 randomly observed individuals do not cover their mouth when sneezing, where the probability of an individual not covering their mouth is given as \( p = 0.267 \).

This scenario follows a **binomial distribution** because there are a fixed number of independent trials (10 people), each with two possible outcomes (cover or not cover their mouth), and the probability of success (not covering the mouth) remains constant.

The probability mass function (PMF) for a binomial distribution is:

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

Where:
- \( n = 10 \) is the number of trials (individuals),
- \( k = 6 \) is the number of successes (people who do not cover their mouth),
- \( p = 0.267 \) is the probability of success,
- \( 1 - p = 0.733 \) is the probability of failure (covering the mouth).

The binomial coefficient \( \binom{n}{k} \) is calculated as:

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

Now, let's calculate the probability \( P(X = 6) \).

\[
P(X = 6) = \binom{10}{6} (0.267)^6 (0.733)^{10-6}
\]

First, calculate the binomial coefficient \( \binom{10}{6} \):

\[
\binom{10}{6} = \frac{10!}{6!4!} = \frac{10 \times 9 \times 8 \times 7}{4 \times 3 \times 2 \times 1} = 210
\]

Now, calculate \( (0.267)^6 \) and \( (0.733)^4 \).

Finally, multiply everything together to find the probability. I'll calculate this for you.The probability that exactly 6 out of 10 randomly observed individuals do not cover their mouth when sneezing is approximately **0.0220** (rounded to four decimal places).

Would you like any further details or clarification?

Here are 5 related questions for deeper exploration:
1. How would the probability change if you observed 15 individuals instead of 10?
2. What is the probability that fewer than 3 out of 10 individuals do not cover their mouth when sneezing?
3. How does the binomial distribution compare to the normal distribution in this context?
4. How would you calculate the expected number of individuals who don't cover their mouth?
5. What is the probability that more than 8 out of 10 individuals do not cover their mouth?

**Tip:** In a binomial distribution, the mean is given by \( \mu = n \cdot p \) and the variance by \( \sigma^2 = n \cdot p \cdot (1 - p) \).

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​ people's habits as they sneeze. Using the binomial distribution, what is the probability that among 10 randomly observed​ individuals, exactly 6 do not cover their mouth when​ sneezing? (Round to four decimal places.)

This problem involves calculating the probability that exactly 6 out of 10 people do not cover their mouth when sneezing, using the binomial distribution formula. The probability of not covering the mouth is given as 0.267. Learn how to apply the binomial formula to find the result. Suitable for students in Grades 10-12.

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