Two percent of the circuit boards manufactured by a particular company are defective. If circuit boards are randomly selected for testing, the probability that the number of circuit boards inspected until a defective board is found is greater than 10 is
(a)1.024 X 10^7
(b) 5.12 X 10^7
(c) 0.1829
(d) 0.8171
(e) None of these
please explain how
To find the probability that the number of circuit boards inspected until a defective board is found is greater than 10, we need to use the concept of geometric distribution.
Let's start by understanding the geometric distribution. In this scenario, the probability of success (finding a defective board) is given as 2% or 0.02. The probability of failure (not finding a defective board) is the complement of the success probability, which is 1 - 0.02 = 0.98.
The geometric distribution models the number of trials required until the first success occurs. In our case, it represents the number of circuit boards inspected until a defective board is found.
The probability that the first defective board is found on the n-th inspection is given by the formula:
P(X = n) = (0.98)^(n-1) * (0.02),
Where X represents the random variable denoting the number of inspections until a defective board is found.
Now, let's calculate the probability that the number of circuit boards inspected until a defective board is found is greater than 10. We need to sum up the probabilities for n greater than 10:
P(X > 10) = P(X = 11) + P(X = 12) + P(X = 13) + ...
Using the formula mentioned earlier, we can calculate each individual probability and sum them up. However, given the options provided, it seems more feasible to approximate the result.
The probability of finding a defective board on the 11th inspection would be:
P(X = 11) = (0.98)^(11-1) * (0.02) ≈ 0.0176733.
The probability of finding a defective board on the 12th inspection would be:
P(X = 12) = (0.98)^(12-1) * (0.02) ≈ 0.0173948.
Applying the same calculation for each subsequent inspection and summing the probabilities:
P(X > 10) ≈ 0.0176733 + 0.0173948 + 0.0171228 + ...
Based on the options provided, the answer closest to this probability is option (d) 0.8171.
Therefore, the answer to the question is option (d) 0.8171.