A company ships computer components in boxes that contain 20 items. Assume that the probability of a defective computer component is 0.2. Find the probability that the first defect is found in the seventh component tested. Round your answer to four decimal places.
(.8)^6 (.2) = 0.0524288
120
To find the probability that the first defect is found in the seventh component tested, we can use the geometric distribution formula:
P(X=k) = (1-p)^(k-1) * p
Where:
- P(X=k) is the probability of the first defect being found at the k-th test.
- p is the probability of a defective computer component (0.2 in this case).
- k is the number of tests (in this case, 7).
Let's plug in the values and calculate the probability:
P(X=7) = (1-0.2)^(7-1) * 0.2
P(X=7) = 0.8^6 * 0.2
P(X=7) ≈ 0.0262
Therefore, the probability that the first defect is found in the seventh component tested is approximately 0.0262 (rounded to four decimal places).
To find the probability that the first defect is found in the seventh component tested, we can use the geometric distribution formula.
The formula for the geometric distribution is:
P(X = k) = (1 - p)^(k-1) * p
Where:
P(X = k) is the probability that the first success occurs on the kth trial,
p is the probability of success on a single trial.
In this case, the probability of finding a defective component is 0.2, so p = 0.2.
We want to find P(X = 7), which is the probability of finding the first defect in the seventh component tested.
Using the geometric distribution formula:
P(X = 7) = (1 - 0.2)^(7-1) * 0.2
= 0.8^6 * 0.2
= 0.262144 * 0.2
= 0.0524
Therefore, the probability that the first defect is found in the seventh component tested is 0.0524, rounded to four decimal places.