# Math Modeling : Discrete Probability Models

## Example Questions

### Example Question #1 : Math Modeling

A beauty supply company manufactures a variety of brushes. Quality control engineers work to ensure that the defected brushes in the factory will be detected prior to shipping them. It is estimated that approximately 0.2% of the brushes made will be defected. Tests can be done individually on the brushes or on batches of the brushes. If the test on a batch of brushes fails, it means that one or more of the brushes in that particular batch are defected. The estimated cost is 4 cents for a single brush, and  cents for a group of  brushes. If a batch fails then all brushes in that batch must be tested individually. Find the most cost-effected quality control procedure for detecting defected brushes.

By testing brush batches in groups of 20 will reduce testing costs without sacrificing the quality.

By testing brush batches in groups of 15 will reduce testing costs without sacrificing the quality.

By testing brush batches in groups of 9 will reduce testing costs without sacrificing the quality.

By testing brush batches in groups of 10 will reduce testing costs without sacrificing the quality.

By testing brush batches in groups of 12 will reduce testing costs without sacrificing the quality.

By testing brush batches in groups of 15 will reduce testing costs without sacrificing the quality.

Explanation:

First identify the known variables and assumptions.

If

If a batch of brushes is tested and if the test shows that all the brushes are good then,

If the batch test shows that there is a defected brush in the batch then,

Use a discrete probability model to find the most cost-effected quality control procedure for detecting defected brushes.

Consider the random variable

that has a probability of

If the probability of a brush being good is  then the probability of a brush being defected is . Then the average expected value of  is as follows:

Now there are  brushes and the probability that one brush is defected is  thus assuming independence, the probability of all  brushes in one test group are good is .

Therefore the expected value of the random variable  is,

Therefore the average testing cost is,

Using the law of large numbers minimizing  results in