A company manufactures tires at a cost of $40 each. The following are probabilities of defective tires in a given production run: probability of no defective tires: 10%; probability of 1% defective tires: 30%; probability of 2% defective tires: 40%; probability of 3% defective tires: 20%. When a defective tire is detected, it must be removed from the assembly line and replaced. This process adds an extra $5 to the cost of the replaced tire. What is the expected cost to the company of a batch of 3000 tires?

The expected cost of the company for a 3000 tires batch is $120255

Step-by-step explanation:

Recall that given a probability of defective tires p, we can model the number of defective tires as a binomial random variable. For 3000 tires, if we have a probability p of having a defective tire, the expected number of defective tires  is 3000p.

Let X be the number of defective tires. We can use the total expectation theorem, as follows: if there are events that partition the whole sample space, and we have a random variable X over the sample space, then

.

So, in this case, we have the following

.

Let Y be the number non defective tires. then X+Y = 3000. So Y = 3000-X. Then E(Y) = 3000-E(X). Then, E(Y) = 2949.

Finally, note that the cost of the batch would be 40Y+45X. Then