A pharmaceutical company receives large shipments of aspirin tablets. The acceptance sampling plan is to randomly select and test 39 ​tablets, then accept the whole batch if there is only one or none that​ doesn't meet the required specifications. If one shipment of 7000 aspirin tablets actually has a 6​% rate of​ defects, what is the probability that this whole shipment will be​ accepted? Will almost all such shipments be​ accepted, or will many be​ rejected?
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Part 1
The probability that this whole shipment will be accepted is 
  
enter your response here.
​(Round to four decimal places as​ needed.)

To solve this problem, we can model it using the binomial distribution. The binomial distribution describes the number of successes in a fixed number of independent Bernoulli trials, where each trial has two possible outcomes: success (a defective tablet) or failure (a non-defective tablet).

Given:
- The number of trials (tablets tested) is \( n = 39 \).
- The probability of a defective tablet is \( p = 0.06 \).
- The number of defectives allowed for the shipment to be accepted is at most 1, meaning either 0 or 1 defective tablets.

We want to calculate the probability of getting 0 or 1 defective tablets from the 39 tested, using the binomial probability formula:

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

Where:
- \( P(X = k) \) is the probability of getting exactly \( k \) defective tablets.
- \( n = 39 \), the number of trials (tablets tested).
- \( p = 0.06 \), the probability of a defective tablet.
- \( \binom{n}{k} \) is the binomial coefficient, which counts the number of ways to choose \( k \) defective tablets from \( n \) trials.

We need to find:

\[
P(\text{accepted}) = P(X = 0) + P(X = 1)
\]

Now, let's calculate this probability.The probability that the whole shipment will be accepted is approximately \( 0.3124 \) (rounded to four decimal places).

This means that about 31.24% of shipments will be accepted if they have a 6% defect rate. Since this acceptance rate is relatively low, many shipments will likely be rejected.

Would you like more details on any part of this explanation, or do you have further questions?

Here are some related questions to expand on this topic:
1. How does the probability change if the defect rate decreases to 3%?
2. What happens to the acceptance probability if the sample size is increased to 50 tablets?
3. How does the acceptance probability change if the threshold is increased to allow 2 defective tablets?
4. What is the probability of rejecting a shipment with the current parameters?
5. How would the acceptance probability differ if the defect rate was 10%?

**Tip**: Understanding the behavior of binomial distributions can help in optimizing sampling plans to balance quality control and efficiency.

A pharmaceutical company receives large shipments of aspirin tablets. The acceptance sampling plan is to randomly select and test 39 tablets, then accept the whole batch if there is only one or none that doesn't meet the required specifications. If one shipment of 7000 aspirin tablets actually has a 6% rate of defects, what is the probability that this whole shipment will be accepted? Will almost all such shipments be accepted, or will many be rejected?

Calculate the probability of accepting a shipment of aspirin tablets with a 6% defect rate using binomial distribution. Understand the statistical concepts, key formulas, and theorems involved in this detailed step-by-step solution. Suitable for college and university students.

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