Acceptance Sampling. Exercises 35 and 36 involve the method of acceptance sampling, whereby a shipment of a large number of items is accepted based on test results from a sample of the items.

Aspirin The MedAssist Pharmaceutical Company receives large shipments of aspirin tablets and uses this acceptance sampling plan: Randomly select and test 40 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 3% 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?

It is given that a shipment is accepted if only one or none of the tablets in a sample of 40 tablets is defective.

Let Xdenote the number of defective tablets.

Success is defined as getting a defective tablet.

The probability of success is computed below:

$$\begin{gathered} p=3\% \\ =\frac{3}{100} \\ =0.03 \end{gathered}$$

The probability of failure is computed below:

$$\begin{gathered} q=1-p \\ =1-0.03 \\ =0.07 \end{gathered}$$

The number of trials (n) is equal to 40.

The binomial probability formula used to compute the given probability is as follows:

$P\left(X=x\right)={}^n{C}_x{\left(p\right)}^x{\left(q\right)}^{n-x}$

Using the binomial probability formula, the probability that one or none of the tablets are defective is computed below:

$$\begin{gathered} P\left(X=0\right)+P\left(X=1\right)={}^{40}{C}_0{\left(0.03\right)}^0{\left(0.07\right)}^{40-0}+{}^{40}{C}_1{\left(0.03\right)}^1{\left(0.07\right)}^{40-1} \\ =0.2957+0.3658 \\ =0.6615 \\ \approx 0.662 \end{gathered}$$

Thus, the probability that the shipment will be accepted is equal to 0.662.

The probability of 0.662 suggests that 66% of such shipments will be accepted, and 33% of the shipments will be rejected.

Since the probability of rejection of shipment is high, the supplier should consider improving the quality of tablets.