Question

One of your suppliers has belatedly realized that about 10% of the batches of a particular component recently supplied to you have a manufacturing fault that has reduced their reliability. There is no external or visual means of identifying these substandard components. Batch identity has, however, been maintained, so your problem is to sort batches that have this fault ('bad' batches) from the rest ('good' batches). An accelerated test has been devised such that components from good batches have a failure probability of 0.02 whereas those from bad batches have a failure probability of 0.2. A sampling plan has been devised as follows:

1 Take a random sample of 25 items from each unknown batch, and subject them to the test.

2 If there are 0 or 1 failed components, decide that the batch is a good one.

3 If there are two or more failures, decide that the batch is a bad one.

There are risks in this procedure. In particular, there are (i) the risk of deciding that a good batch is bad: and (ii) the risk of deciding that a bad batch is a good. Use Bayes' theorem and your answers to question 2 and 3 to evaluate these risks.

Answer 1

Let good batch be denoted by G, bad batch be denoted by B, failure in test be denoted as F

Based on the information above, we know that,

P(F|G) = 0.02, P(F|B) =0.2

Sample size, n = 25

(i) To calculate Risk of deciding that a good batch is bad, we first need to calculate P(0 failures), P(1 failure) and P(>1 failure) in the good sample of 25

Using Binomial distribution to calculate probability of 0 failures:-

Here, p = 0.02 (good batch), k=0 , n =25

Using the general binomial probability formula,

P(k out of n) =

Putting the values from above, we get

= 0.603

Similarly, calculating the probability of 1 failure:

Here, p = 0.02 (good batch), k=1 , n =25

Using the general binomial probability formula,

P(k out of n) =

Putting the values from above, we get

P(1 failure|good batch) = 0.308

Hence, P(0 or 1 failure|good batch) = 0.603 + 0.308 = 0.911

Thus, P(>1 failure|good batch) = 1 - P(0 or 1 failure|good batch) = 1-0.911 = 0.089

Thus, P(decided bad batch|Good batch) = 0.089 = Risk of deciding good batch as bad

(ii)

To calculate Risk of deciding that a bad batch is good, we first need to calculate P(0 failures) and P(1 failure) a bad sample of 25

Using Binomial distribution to calculate probability of 0 failures:-

Here, p = 0.2 (bad batch), k=0 , n =25

Using the general binomial probability formula,

P(k out of n) =

Putting the values from above, we get

= 0.0038

Similarly, calculating the probability of 1 failure:

Here, p = 0.2 (bad batch), k=1 , n =25

Using the general binomial probability formula,

P(k out of n) =

Putting the values from above, we get

P(1 failure|bad batch) = 0.0236

Hence, P(0 or 1 failure|bad batch) = 0.0038 + 0.0236 = 0.0274

Thus, P(<=1 failure|bad batch) =0.0274

Thus, P(decided good batch|Bad batch) = 0.027 = Risk of deciding good batch as bad