https://file.aiquickdraw.com/mathbot/file/1729300650024m1gy7xxq.jpg

This problem involves **binomial probability**. Let's go through each part of the question:

### Given Information:
- The probability that a randomly selected individual will **not** cover their mouth when sneezing is \( p = 0.267 \).
- The total number of individuals observed is \( n = 12 \).

The binomial probability formula is:
\[
P(X = k) = \binom{n}{k} \cdot p^k \cdot (1 - p)^{n-k}
\]
Where:
- \( n = 12 \) (total individuals),
- \( p = 0.267 \) (probability of not covering mouth),
- \( k \) is the number of individuals who do not cover their mouth.

### Part (a): Probability that exactly 8 do **not** cover their mouth when sneezing.

For this, we need to calculate the probability of \( X = 8 \), where \( X \) follows a binomial distribution with \( n = 12 \) and \( p = 0.267 \).
\[
P(X = 8) = \binom{12}{8} \cdot (0.267)^8 \cdot (1 - 0.267)^{12 - 8}
\]
From the image, the result provided is **0.0037** (rounded to 4 decimal places).

### Part (b): Probability that fewer than 5 do **not** cover their mouth when sneezing.

Here, we need to calculate the probability that \( X < 5 \), which means summing the probabilities for \( X = 0, 1, 2, 3, 4 \):
\[
P(X < 5) = P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3) + P(X = 4)
\]
The result from the image is **0.8053**.

### Part (c): Would it be surprising if fewer than half of the 12 individuals do not cover their mouth?

"Fewer than half" of 12 individuals would mean fewer than 6 (so, 5 or fewer individuals not covering their mouth). Based on the calculation from part (b), the probability that fewer than 5 do not cover their mouth is **0.8053**, which is quite large, indicating this outcome is not surprising.

We are asked whether this probability is surprising or not based on a threshold probability of **0.05**. Since 0.8053 is much larger than 0.05, **it would not be surprising**.

Would you like more details or further explanation on any of the steps?

### Follow-up Questions:
1. What does the binomial distribution represent in this context?
2. How do you calculate binomial coefficients in binomial probability?
3. How would the probability change if the sample size increased to 20?
4. What would happen if the probability of not covering one's mouth was higher (e.g., 0.4)?
5. How do we interpret the probability threshold of 0.05 in part (c)?

**Tip**: When rounding probabilities, it is important to round to four decimal places (or as specified) to maintain precision in the calculations.

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, calculate the following:

Explore the binomial probability of individuals not covering their mouth when sneezing. Learn how to compute exact probabilities using a given probability of 0.267 and a sample of 12 individuals, and determine whether outcomes are surprising based on a probability threshold of 0.05.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation