Question

A manufacturing company produces 10,000 products a day. From previous experience, quality control analysts estimate that 0.5% of products manufactured have some kind of defect and cannot be sold. The defective products are thrown away and treated as a loss. It costs $1.00 to manufacture, distribute, and market each individual unit, regardless of whether or not the product has a defect. The unit retail price is $5.00, and each of the non-defective products sells with probability 0.95.

6. Calculate the probability that 53 products are defective on any given day. Include the formula for your calculation, though you may evaluate it in R.
7. Calculate the expected value of the number of products that have no defects. Do this by hand, providing the formula and correct calculation.

Answer 1

From the provided information:

The probability that the products are defective on any given day is p=0.05

The number of product produce per day are n=10,000

a) Let X be the number of defective products.

Therefore,

The probability that 53 products are defective on any given day is obtained by:

### By using R command:

> dbinom(53,10000,0.05)
[1] 5.905232e-149

b) The probability of non defective product is q=0.95

Let Y be the nuber of non defective product.

Therefore,

The expected value of the number of products that have no defects is: