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When purchasing bulk orders of batteries, a toy manufacturer uses this acceptance sampling plan: Randomly select and test 42 batteries and determine whether each is within specifications. The entire shipment is accepted if at most 2 batteries do not meet specifications. A shipment contains 7000 batteries, and 2% of them do not meet specifications. What is the probability that this whole shipment will be accepted? Will almost all such shipments be accepted, or will many be rejected? The probability that this whole shipment will be accepted is (Round to four decimal places as needed.) The company will accept % of the shipments and will reject % of the shipments, so (Round to two decimal places as needed.)

When purchasing bulk orders of batteries a toy manufacturer uses this acceptance sampling plan Randomly select and test 42 batteries and determine whether each class=


Answer :

Solution

Here, we would calculate the probability that one is defective, two are defective, three are defective and the probability that none are defective and sum them together.

The probability that any individual battery is not defective = 1 - 2% = 0.98

We need 42 of them,

Therefore, P(that none are defective) = (0.98)^42 = 0.4281

The probability that a specific battery will be the only defective battery is (0.02)*(0.98)^41

Since we have 42 of them, ((0.02)*(0.98)^41)*42 = 0.367

Note that

[tex]42C2=\frac{42!}{(42-2)!2!}=861[/tex]

=> ((0.02)^2*(0.98)^40)*861 = 0.1535

Therefore, 0.4281 + 0.367 + 0.1535 = 0.9486

The probability shows that about 94.86% of all shipments will be accepted

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