A pharmaceutical company receives large shipments of aspirin tablets. The acceptance sampling plan is to randomly select and test 2020 ​tablets, then accept the whole batch if there is only one or none that​ doesn't meet the required specifications. If a particular shipment of thousands of aspirin tablets actually has a 22​% rate of​ defects, what is the probability that this whole shipment will be​ accepted?
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Part 1
The probability that this whole shipment will be accepted is enter your response here.
​(Round to three decimal places as​ needed.)

The scenario describes a binomial probability problem, where we are trying to determine the probability that a shipment of aspirin tablets will be accepted. In this case:

- The number of trials (tablets tested) \( n = 2020 \).
- The probability of success (a defective tablet) \( p = 0.22 \).
- The acceptance criteria is 0 or 1 defective tablet.

The probability of exactly \( k \) defective tablets in \( n \) trials follows the binomial distribution formula:

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

We are interested in the probability that there are 0 or 1 defective tablet, which means we need to calculate:

\[
P(X = 0) + P(X = 1)
\]

### Step 1: Calculate \( P(X = 0) \)
\[
P(X = 0) = \binom{2020}{0} \cdot (0.22)^0 \cdot (0.78)^{2020}
\]
\[
P(X = 0) = 1 \cdot 1 \cdot (0.78)^{2020}
\]

### Step 2: Calculate \( P(X = 1) \)
\[
P(X = 1) = \binom{2020}{1} \cdot (0.22)^1 \cdot (0.78)^{2019}
\]
\[
P(X = 1) = 2020 \cdot 0.22 \cdot (0.78)^{2019}
\]

Let's calculate the values to find the total probability of acceptance.



A pharmaceutical company receives large shipments of aspirin tablets. The acceptance sampling plan is to randomly select and test 2020 tablets, then accept the whole batch if there is only one or none that doesn't meet the required specifications. If a particular shipment of thousands of aspirin tablets actually has a 22% rate of defects, what is the probability that this whole shipment will be accepted?

This problem explores the binomial probability in the context of a pharmaceutical company's acceptance sampling plan. Given that a 22% defect rate is observed in a shipment, we calculate the probability of accepting the shipment by testing 2020 tablets and allowing up to 1 defect. Detailed solution with binomial distribution formulas and key probability concepts.

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