With one method of a procedure called acceptance sampling, a sample of items is randomly selected without replacement and the entire batch is rejected if there is at least one defective unit. The Condor Electronics Company has just manufactured 4500 gamedisks, and 170 are defective. If 4 of these gamedisks are randomly selected and tested, what is the probability that the entire batch will be rejected?

Report the answer as a percent with one decimal place accuracy. (You need not enter the "%" symbol)
Probability =

To solve this problem, we'll calculate the probability that at least one of the four randomly selected gamedisks is defective. If at least one is defective, the entire batch will be rejected.

### Step 1: Calculate the probability of selecting no defective gamedisks

First, we find the probability of selecting 4 non-defective gamedisks. The total number of gamedisks is 4500, and the number of defective ones is 170. The number of non-defective gamedisks is therefore \(4500 - 170 = 4330\).

The probability of selecting the first non-defective gamedisk is:
\[
\frac{4330}{4500}
\]

The probability of selecting the second non-defective gamedisk, given that the first one was non-defective, is:
\[
\frac{4329}{4499}
\]

Similarly, the probability for the third non-defective gamedisk is:
\[
\frac{4328}{4498}
\]

And for the fourth non-defective gamedisk:
\[
\frac{4327}{4497}
\]

The combined probability of selecting 4 non-defective gamedisks is:
\[
P(\text{no defectives}) = \frac{4330}{4500} \times \frac{4329}{4499} \times \frac{4328}{4498} \times \frac{4327}{4497}
\]

### Step 2: Calculate the probability that the batch is rejected

The probability that the batch will be rejected (i.e., at least one defective gamedisk is selected) is the complement of the probability of selecting no defective gamedisks:
\[
P(\text{batch rejected}) = 1 - P(\text{no defectives})
\]

Let's calculate this.

### Step 3: Perform the calculations
We'll calculate the probability step by step.The probability that the entire batch will be rejected is **14.3%**.

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

Here are 5 related questions for further exploration:
1. How would the probability change if 5 gamedisks were selected instead of 4?
2. What is the probability of selecting exactly one defective gamedisk in the sample?
3. How would the result change if the sample was selected with replacement?
4. What if the total number of defective gamedisks was unknown? How could we estimate it?
5. How does this probability compare to the probability of rejecting a smaller batch size?

**Tip:** Acceptance sampling is a useful method in quality control when testing every item is impractical, but it requires a well-understood statistical foundation to set appropriate sample sizes and acceptance criteria.

Calculate the probability of rejecting the entire batch in acceptance sampling, based on randomly testing 4 out of 4500 gamedisks, where 170 are defective. Learn the step-by-step solution with a detailed probability analysis.

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