Question

Suppose that ten items are chosen at random from a large batch delivered to a company. the manufacturer claims that just 4% of the items in the batch are defective. assume that the batch is large enough so that even though the selection is made without replacement, the number 0.04 can be used to approximate the probability that any one of the ten items is defective. in addition, assume that because the items are chosen at random, the outcomes of the choices are mutually independent. finally, assume that the manufacturer's claim is correct. (round your answers to three decimal places.)

(a) What is the probability (as a %) that none of the ten is defective? X

Answer 1

The probability that none of the ten items are defective, given a 4% defect rate and mutually independent selections, is 66.5%.

Step-by-step explanation:

The question is asking for the probability that none out of ten randomly selected items from a large batch are defective, given that the manufacturer claims there is a 4% defect rate. Assuming the defect rate is correct and the selections are mutually independent, we use the binomial probability formula.

To find the probability of getting no defective items (X = 0), we calculate (0.96)^10, which is the probability of an item not being defective raised to the power of the number of items chosen. Using this calculation:

P(X = 0) = (0.96)^10

After calculating, we get:

P(X = 0) = 0.665

To express this as a percentage, we multiply by 100:

Probability (as a %) = 66.5%

So the probability that none of the ten items are defective is 66.5%.