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The problem presented is about finding the probability that a shipment of aspirin will be accepted based on a sampling test. Here’s a breakdown of the key points:

- **Sampling Plan**: 54 tablets are randomly selected and tested from a shipment of 5000 aspirin tablets.
- **Acceptance Criterion**: The shipment will be accepted if only 1 or 0 tablets in the sample are defective.
- **Defect Rate**: The defect rate is given as 2%, or 0.02 probability for each tablet being defective.
- **Question**: What is the probability that the whole shipment will be accepted? (i.e., 1 or fewer defects are found in the sample of 54 tablets).

### Steps to solve the problem:
1. **Model the Situation**: We can model this using a **binomial distribution** because each tablet is either defective or not defective, and each trial (tablet) is independent.

   - \( n = 54 \) (number of tablets sampled)
   - \( p = 0.02 \) (probability of defect for each tablet)
   - The number of defective tablets follows the binomial distribution: \( X \sim \text{Binomial}(n = 54, p = 0.02) \).

2. **Acceptance Probability**: The shipment will be accepted if 0 or 1 defective tablets are found in the sample, so we need to calculate \( P(X \leq 1) \).

3. **Binomial Probability Formula**: 
   \[
   P(X = k) = \binom{n}{k} p^k (1-p)^{n-k}
   \]
   for \( k = 0 \) and \( k = 1 \), and sum the probabilities.

4. **Calculations**:
   - \( P(X = 0) \): Probability of 0 defective tablets.
   - \( P(X = 1) \): Probability of 1 defective tablet.

   We sum these probabilities to find \( P(X \leq 1) \).

Let me calculate the probability.The probability that the whole shipment will be accepted (i.e., that 0 or 1 defective tablets are found in the sample of 54) is approximately **0.7061** (rounded to four decimal places). This means there is about a 70.61% chance that the shipment will be accepted.

Do you have any questions, or would you like more details?

Here are 5 related questions to deepen your understanding:
1. How would the probability change if the defect rate were 5% instead of 2%?
2. What would happen to the acceptance probability if the sample size were increased to 100 tablets?
3. Can you describe the binomial distribution and how it applies to quality control sampling?
4. What would the probability of acceptance be if the company allowed 2 defective tablets in the sample?
5. How does increasing the sample size generally affect the likelihood of acceptance?

**Tip**: In binomial distributions, increasing the sample size while keeping the defect rate constant typically makes it easier to detect defects.

A pharmaceutical company receives large shipments of aspirin tablets. The acceptance sampling plan is to randomly select and test 54 tablets, then accept the whole batch if there is only one or none that doesn't meet the required specifications. If one shipment of 5000 aspirin tablets actually has a 2% rate of defects, what is the probability that this whole shipment will be accepted?

Determine the probability of accepting a shipment of 5000 aspirin tablets with a 2% defect rate using a binomial distribution, testing 54 tablets, and allowing 1 or fewer defects. Learn how to calculate this using the binomial probability formula.

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