Calculating the Probability of a Sample Containing Bad Parts

Calculating the Probability of a Sample Containing Bad Parts

Received a question from a reader this morning that will make a nice tutorial.

A box contains 27 black and 3 red balls. A random sample of 5 balls is drawn without replacement. What is the probability that the sample contains one red ball?

So here’s my thinking and two ways to solve this problem. Instead of red and black balls in an urn type problem, which is pretty abstract, let’s say we know 3 bad parts are in a bin of 30 total parts.

We need five parts to build a system and if one is bad, we have to repair the system, which has a cost.

Or, should we sort and test each part which also has a cost and takes time to accomplish?

The details will be in the cost of inspections and the cost of a system repair, yet we also need to know the probability of building a system with one bad part out of the five needed.

Probability and Combinations

My first thought in solving the problem was the nature of the sampling, without replacement. This is realistic in that we need five parts for the build, so selecting one and putting it back in the bin would serve no…