Find the probability.
-In a batch of 8,000 clock radios 6% are defective.A sample of 9 clock radios is randomly selected without replacement from the 8,000 and tested.The entire batch will be rejected if at least one of those tested is defective.What is the probability that the entire batch will be rejected?
A)0.573
B)0.111
C)0.0600
D)0.427

Question

Quizplus · Accepted Answer

The probability of selecting a non-defective clock radio on the first test is 0.94 (1 - 0.06). The probability of selecting a non-defective clock radio on the second test is 0.933 (the batch is without replacement, so there is one fewer non-defective clock radio in the batch). Continue this process until the probability of selecting a non-defective clock radio on the ninth test is 0.88 (there are only 7992 clock radios left in the batch at this point).

To find the probability that at least one clock radio in the sample is defective, we can find the probability that all of them are non-defective and subtract that from 1:
P(at least one defective) = 1 - P(all non-defective)

P(all non-defective) = 0.94 * 0.933 * ... * 0.88
P(all non-defective) = 0.511

Therefore,
P(at least one defective) = 1 - P(all non-defective)
P(at least one defective) = 1 - 0.511
P(at least one defective) = 0.489

So the probability that the entire batch will be rejected is the same as the probability that at least one clock radio in the sample is defective, which is 0.489. Subtract this from 1 to get the probability that the entire batch will not be rejected:
P(batch not rejected) = 1 - 0.489
P(batch not rejected) = 0.511

Therefore, the probability that the entire batch will be rejected is:
P(batch rejected) = 1 - P(batch not rejected)
P(batch rejected) = 1 - 0.511
P(batch rejected) = 0.489, which is closest to choice D (0.427).

Find the Probability. -In a Batch of 8,000 Clock Radios 6% Are Defective.A