A manufacturer must supply a customer with 100 items. Typically 2% of the itms a

Question

A manufacturer must supply a customer with 100 items. Typically 2% of the itms are identified as defective and the items can be assumed to be independent.
(a) If exactly 100 items are shipped to a customer, what is the probability that all items
are good, and the customer will be satisfied?
(b) If exactly 102 items are shipped to a customer who is only billed for 100 items, what
is the probability that receiving at least 100 good items will satisfy the customer?
(c) If exactly 105 items are shipped to a customer who is only billed for 100 items, what is the probability that receiving at least 100 good items will satisfy the customer?
(d) What are the mean and variance of the number of faulty ones sent in each case?

Explanation / Answer

(a) P(100 not defective) = P(1st not defective AND 2nd not defective AND ... 100 not defective)    Whenever you see AND in a problem, you are to MULTIPLY the probabilities
P(100 not defective) = (0.98)(0.98)(0.98)...(0.98) = (0.98)100

(b) P(100 not defective OR 101 not defective OR 102 not defective) Whenever you see OR in a problem, you are to ADD the probabilities (0.98)100+(0.98)101+(0.98)102
(c) Same as part (b):    (0.98)100 + (0.98)101 + (0.98)102 + (0.98)103 + (0.98)104 + (0.98)105

I'm assuming that you haven't yet learned binomial probabilities since it is the beginning of the semester. However, if you have then you can answer these questions a little bit differently.

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