Expected Profit for Binomial Variable

Question

Part (a) I am familiar with:

(a) P(batch is rejected) = P(X greater than or equal to 3)

and n = 15 and p(defective) = 0.1

This gives me the correct answer of 0.1841

I am stuck at part 2! I have no idea how to tackle this

The profit for B good batches = 38b+5(1-b)-20B = 23b-5 where b is the number of GOOD batches.

The profit per batch = (23b-5)/b

To find the expected number of GOOD batches, I must find the probability that a batch is good * number of trials.

I do know that

E(x) = n*p

but what is n, and what is p?

The P(batch is good) = 1-0.1841=0.8159 N = Number of trials, that is number of batches on shelf... but how many is this going to be?

Answer 1

The expected profit per batch is $0.8159\times38+0.1841\times5-20$ in dollars. Where $0.8159$ stands for the probability that a batch is good and $0.1841$ for the probability that a batch is not good.

Answer 2

Profit = Revenue - Cost. The cost is constant, 20. Now you need to find the revenue (expectation): $$ \mathbf{E} X = 38 \cdot P(good) + 5 \cdot P(Bad) $$ You have done most of the remaining part. The probability the batch is good is found using negative binomial distribution.