https://file.aiquickdraw.com/mathbot/file/17286052361517zb6emw6.jpg

The problem asks to determine the probability \( P(r \leq 1 \text{ defective}) \) from a sample of size \( n = 15 \), given that 10% of the population is defective. This is a binomial probability problem, where we use the binomial distribution formula:

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

Where:
- \( n = 15 \) is the sample size,
- \( p = 0.10 \) is the probability of a defect,
- \( k \) is the number of defective items,
- \( \binom{n}{k} \) is the binomial coefficient.

We are tasked with finding \( P(r \leq 1 \text{ defective}) \), which means the probability of finding 0 or 1 defective items. This is the cumulative probability for \( r = 0 \) and \( r = 1 \):

\[
P(r \leq 1) = P(r = 0) + P(r = 1)
\]

### 1. For \( r = 0 \):
\[
P(r = 0) = \binom{15}{0} (0.10)^0 (0.90)^{15} = (1) \cdot 1 \cdot (0.90)^{15} \approx 0.2059
\]

### 2. For \( r = 1 \):
\[
P(r = 1) = \binom{15}{1} (0.10)^1 (0.90)^{14} = 15 \cdot 0.10 \cdot (0.90)^{14} \approx 15 \cdot 0.10 \cdot 0.2287 = 0.3431
\]

### Total Probability:
\[
P(r \leq 1) = 0.2059 + 0.3431 = 0.5490
\]

So, the probability that there are 1 or fewer defective items in a sample of 15 is approximately 0.549, or 54.9%.

Would you like more details on any of the steps or concepts involved?

Here are 5 related questions:
1. What is the probability of getting exactly 2 defective items in a sample of 15?
2. How does the sample size affect the probability in a binomial distribution?
3. Can we use the normal approximation for this problem?
4. How would the probability change if the defect rate was 5% instead of 10%?
5. What is the cumulative probability for \( P(r \leq 3) \)?

**Tip:** When dealing with binomial probabilities, the cumulative probability helps to find the likelihood of a range of outcomes rather than just one specific outcome.

If a certain population is known to be 10% defective, determine the P(r <= 1 defective) from a sample of n = 15.

Calculate the probability of 1 or fewer defective items in a sample of 15 when the population defect rate is 10%. This problem applies the binomial distribution formula and is suitable for advanced high school or college-level students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation