Circuit boards for wireless telephones are etched, in an acid bath, in batches of 100 boards.A sample of seven boards is randomly selected from each lot for inspection.A batch contains two defective boards; and x is the number of defective boards in the sample.P(x=1)is _______.
A)0.1315
B)0.8642
C)0.0042
D)0.6134
E)0.6789

Question

Quizplus · Accepted Answer

The probability P(x=1) can be calculated using the hypergeometric distribution formula, which is appropriate for sampling without replacement from a finite population. The formula for finding the probability of getting exactly k successes (defective boards in this case) in n draws (sample size) from a population of N containing K successes (defective boards in the batch) is:$$ P(x=k) = \frac{{\binom{K}{k} \cdot \binom{N-K}{n-k}}}{{\binom{N}{n}}} $$Given values are N=100 (total boards), K=2 (defective boards), n=7 (sample size), and k=1 (number of defective boards in the sample). Plugging these values into the formula gives:$$ P(x=1) = \frac{{\binom{2}{1} \cdot \binom{98}{6}}}{{\binom{100}{7}}} $$$$ P(x=1) = \frac{{2 \cdot 1221759}}{{16007560800}} \approx 0.1315 $$Therefore, the correct answer is A) 0.1315, which represents the probability of finding exactly one defective board in the sample.

Circuit Boards for Wireless Telephones Are Etched, in an Acid