With one method of a procedure called acceptance sampling, a sample of items is randomly selected without replacement and the entire batch is accepted if every item in the sample is okay. The ABC Electronics Company has just manufactured 1650 write-rewrite CDs, and 160 are defective. If 5 of these CDs are randomly selected for testing, what is the probability that the entire batch will be accepted?

Rounded to four decimal place accuracy.

To solve this, we'll use the concept of hypergeometric probability since the selection is made without replacement.

### Key information:
- Total number of CDs: \( N = 1650 \)
- Number of defective CDs: \( D = 160 \)
- Number of non-defective CDs: \( N - D = 1650 - 160 = 1490 \)
- Sample size: \( n = 5 \)

The entire batch will be accepted if none of the 5 CDs selected is defective. This means we are choosing 5 non-defective CDs from the total pool of 1490 non-defective CDs.

The probability of selecting exactly \( k = 5 \) non-defective CDs in the sample can be found using the **hypergeometric probability formula**:

\[
P(\text{accepted}) = \frac{\binom{1490}{5}}{\binom{1650}{5}}
\]

Where \( \binom{a}{b} \) is the combination function (i.e., "a choose b").

### Steps:
1. Calculate \( \binom{1490}{5} \), the number of ways to choose 5 non-defective CDs.
2. Calculate \( \binom{1650}{5} \), the number of ways to choose 5 CDs out of 1650.
3. Divide these two values to get the probability.

Let’s compute this.The probability that the entire batch will be accepted is approximately \( 0.6001 \), or 60.01%.

Would you like more details on the calculation or have any questions?

Here are some related questions for deeper exploration:
1. What happens to the probability if the sample size is increased to 10?
2. How would the probability change if the number of defective CDs increased to 200?
3. What is the probability of finding at least one defective CD in a sample of 5?
4. How would this problem change if the selection was with replacement?
5. How does hypergeometric probability differ from binomial probability?

**Tip:** When sampling without replacement, the hypergeometric distribution is typically used, especially when the population is not large enough to assume independence (as in binomial distribution).

With one method of a procedure called acceptance sampling, a sample of items is randomly selected without replacement and the entire batch is accepted if every item in the sample is okay. The ABC Electronics Company has just manufactured 1650 write-rewrite CDs, and 160 are defective. If 5 of these CDs are randomly selected for testing, what is the probability that the entire batch will be accepted?

Calculate the probability that a batch of 1650 CDs from ABC Electronics, with 160 defective units, will be accepted based on acceptance sampling. Learn how to apply hypergeometric probability, key combinatoric principles, and understand real-world applications of probability in quality control.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation