Each day, Monday through Friday, a batch of components sent by a first supplier arrives at a certain inspection facility. Two days a week, a batch also arrives from second supplier.

This is an excerpt from a problem in “Probability and Statistics for Engineering and the Sciences” by J.L. Devore.

How do I find the probability that two batches arrive in a randomly selected day? How about one batch? Or no batch? In the textbook, it’s given that the probability of one batch arriving is 0.6 and two batch arriving is 0.6. But together it means that at least one batch arrives in a randomly selected day.

However, it can be seen that there is a probability that **no** batches arrive in a day. For example, in a particular Sunday, the first supplier (who only sends a batch monday through friday) will not a send. The second supplier, who sends them two days a week, may choose not to send it on a Sunday.

Here’s the full question and the solution:

Each day, Monday through Friday, a batch of components sent by a first supplierarrives at a certain inspection facility. Two days a week, a batch also arrives froma second supplier. Eighty percent of all supplier 1’s batches pass inspection, and90% of supplier 2’s do likewise. What is the probability that, on a randomlyselected day, two batches pass inspection? We will answer this assuming thaton days when two batches are tested, whether the first batch passes is independ-ent of whether the second batch does so.