the quality control inspector of a production plant will reject a batch of syringes it two or more defective syringes are found in a random sample of ten syringes taken from the batch. suppose the batch contains 2% defective syringes. What is the probability that the batch will be accepted?
The easiest way to do this is to use a binomial probability table. You would need to find P(0) and P(1), add the probabilities together, then subtract that value from 1.
To find the probability that the batch will be accepted, we need to calculate the probability of finding less than two defective syringes in a random sample of ten syringes from the batch.
First, let's find the probability of finding exactly zero defective syringes in the sample.
The probability of selecting a defective syringe from the batch is given as 2%. Therefore, the probability of selecting a non-defective syringe is 1 - 0.02 = 0.98.
The probability of selecting zero defective syringes in a sample of ten can be calculated using the binomial probability formula:
P(X=k) = (n choose k) * p^k * (1-p)^(n-k)
where:
- n is the number of trials (in this case, selecting 10 syringes from the batch)
- k is the number of successful trials (in this case, zero defective syringes)
- p is the probability of success (in this case, the probability of selecting a non-defective syringe)
Using the formula, we can calculate the probability of finding exactly zero defective syringes:
P(X=0) = (10 choose 0) * 0.98^0 * (1-0.98)^(10-0)
= 1 * 1 * 0.98^10
≈ 0.8163
Next, let's find the probability of finding exactly one defective syringe in the sample.
P(X=1) = (10 choose 1) * 0.98^1 * (1-0.98)^(10-1)
= 10 * 0.98 * 0.02^9
≈ 0.1837
Finally, to find the probability that the batch will be accepted, we need to sum the probabilities of finding zero or one defective syringes:
P(Batch accepted) = P(X=0) + P(X=1)
≈ 0.8163 + 0.1837
≈ 1
Therefore, the probability that the batch will be accepted is approximately 1, or 100%.