Question

A company purchases electronic components in batches of 100 and the supplier guarantees that

there will be no more than 5 defective components in each batch. Before acceptance of a particular
batch the company has a policy of selecting without replacement two components for testing. If both
components are satisfactory the batch is accepted and if both are defective the batch is rejected.
However, if only one is defective another component is selected and if this is satisfactory the batch
is accepted while if it is defective the batch is rejected. If the probability that a component is
defective is 5% what is the probability that the batch will be accepted?

Answer 1

To solve this problem, we can use conditional probability.

First, let's find the probability that both components selected for testing are satisfactory. Since we're selecting without replacement, the probability of both being satisfactory is (95% * 94%) = 90.3%.

Next, let's find the probability that both components selected for testing are defective. This is (5% * 4%) = 0.2%.

Now, let's find the probability that one component is defective and the other is satisfactory. We can find this by multiplying the probability of one component being defective (5%) by the probability of the other component being satisfactory (94%). This gives us (5% * 94%) = 4.7%.

Finally, we'll find the probability that the batch will be accepted. The batch will be accepted if both components are satisfactory or if only one component is defective and the other component selected is satisfactory. The probability of this is 90.3% + 4.7% = 95%.

So the probability that the batch will be accepted is 95%.

Answer 2

To calculate the probability that the batch will be accepted, we need to consider the different scenarios. If both components tested are satisfactory, the batch is accepted. If both components tested are defective, the batch is rejected. If one component is defective and the other is satisfactory, we need to test an additional component. The batch is accepted if the additional component is satisfactory.

Step-by-step explanation:

To calculate the probability that the batch will be accepted, we need to consider the different scenarios.

If both components tested are satisfactory, the batch is accepted with probability 1.

If both components tested are defective, the batch is rejected with probability 1.

If one component is defective and the other is satisfactory, we need to test an additional component. The probability that the additional component is satisfactory is 0.95, and the probability that it is defective is 0.05. Hence, the batch is accepted with probability 0.95.

Therefore, to calculate the overall probability of accepting the batch, we need to consider the probabilities of each scenario happening. Since each scenario is mutually exclusive, we can sum up the probabilities to get the final answer.

P(batch accepted) = P(both satisfactory) + P(both defective) + P(one defective and one satisfactory) * P(additional component is satisfactory)