Suppose a lot of 100 items manufactured in an industrial plant actually contains items that do not meet either the manufacturer’s or the buyer’s specifications. Let us denote these items by calling them D items while the remainder of the manufacturer’s output, those items that do meet the manufacturer’s and the buyer’s specifications, we will call G items.

Now the manufacturer wishes to inspect a random sample of the items produced by the production line. It may be that the inspection process destroys the product or that the inspection process is very costly, so the manufacturer uses sampling and so inspects only a portion of the manufactured items.

As an example, suppose the lot of 100 items actually contains 10 D items and 90 G items and that we select a random sample of 5 items from the entire lot produced by the manufacturer.

There are  possible samples. The TI83 calculates this as 75,287,520. Suppose we want the probability that the sample contains exactly 3 of the D items. Since we assume that each of the samples is equally likely, this probability is

making it fairly unlikely that this sample will find three of the items that do not meet specifications.

It may be of interest to find the probabilities for all the possible values of D. This is often called the probability distribution of the random variable D. That is, we want to find the values of the function

  for d = 0,1,2,3,4,5.

To produce a graph of these probabilities, store the values of d in a list and then store the value of the function above in another list as follows.

  seq (X,X,0,5) -> L₁

  seq ((5 _nC_r X * 90 _nC_r (5 - X))/100 _nC_r 5,X,0,5) -> L₂

Then a STAT PLOT will exhibit the graph which will resemble the graph shown below (but this graph was not made with the TI83).

What should the manufacturer do if items not meeting specifications are discovered in the sample? Normally, one of two courses was followed: either the D items found in the sample are replaced by G items or the entire lot is inspected and any D items found in the entire lot are replaced by G items. The last course is followed usually if the sample does not exhibit too many D items, and, of course, can only be followed if the sampling is not destructive.

If the sample does not contain too many D items, the lot is accepted and sent to the buyer, perhaps after some D items in the sample are replaced by G items. Otherwise, the lot is rejected. Hence the process is called acceptance sampling.

We will explore the second possibility noted above here, namely that if any D items at all are found in the sample then the entire lot is inspected and any D items in it are replaced with G items. So, the entire delivered lot consist of G items when the sample detects any D items at all.

This clearly will improve the quality of the lot of items sold, but it is not clear how much of an improvement will result. The process has some surprising consequences and we will now explore this procedure.

To be specific, let us suppose that the lot is accepted only if the sample contains no D items whatsoever. Let us also assume that we do not know how many D items are in the lot, so we will suppose that there are d of these in the lot.

The lot is then accepted with probability

This is a decreasing function of d; the larger d is the more likely the sample will contain some D items and hence the lot will not be accepted. A graph of this function will resemble the graph below.

Finally we consider the average percentage of D items delivered to the customer with this acceptance sampling plan. This is often called the average outgoing quality in the quality control literature.

Recall that the average of a quantity is found by multiplying the values of that quantity by the probability of that quantity and adding the results. So if a random variable is D, whose specific values are d, then the average value of D is

Here we wish to find the average value of the percentage of D items delivered to the buyer, or the average of the quantity . This is the average outgoing quality or AOQ or 

But we have a very special circumstance here. The delivered lot has percentage D items of  only if the sample contains no D items whatsoever; otherwise, the lot has 0% D items due to the replacement plan. So the average outgoing quality is

     
or
     

A graph of this function will resemble the graph below:

We notice that the graph attains a maximum value; this may not have been anticipate! This means that, regardless of the quality of the lot, there is a maximum for the average percentage of D items that can be delivered to the customer! This maximum can be found using the TI83 and the graph above. Here are the values of the AOQ near the maximum value:
 
 

so we see that the maximum AOQ occurs when d = 16, so the maximum average percentage of D items that can be delivered to the customer is 0.066!

Sampling here has had a dramatic impact on the average percentage of D items delivered to the customer.

This is just one example of how probability and statistics can assist in delivering high quality product to consumers. There are many other techniques used which are called in general statistical quality control methods, or SPC; these have found wide use in industry today.