Question

A. Suppose, that 1000 items are selected for two quality control tests (dependent tests), say Test 1 and Test 2. Suppose further, that the probability that a randomly selected product passes Test 1 is 0.8, and the probability that the product passes Test 2 is 0.7. It is further found that the probability that the product passes at least one of the tests, either Test 1 or Test 2, or both Tests 1 and 2, is 0.95. Find the probability that the product passes both tests. Define the events, develop probability expressions and solve.

B. Consider a situation similar to part (a) above with the difference that now the quality control tests, Test 1 and Test 2, are considered to be independent tests. Suppose further that the probability that the product passes Test 1 is 0.8, and the probability that the products passes Test 2 is 0.9.

B.1 What is the probability that a randomly selected item passes both tests. Define events, develop probability expressions and solve.

B.2 What is the probability the product passes at least one test, (either Test 1 or Test 2 or both Tests 1 and 2). Develop probability expressions and solve.

Answer 1

A) P(passes at least one of the tests) = P(pass test 1) + P(pass test 2) - P(pass both tests)

Or, 0.95 = 0.8 + 0.7 - P(pass both tests)

Or, P(pass both tests) = 0.8 + 0.7 - 0.95

Or, P(pass both tests) = 0.55

B. 1) P(pass both tests) = P(pass test 1) * P(pass test 2)

= 0.8 * 0.9 = 0.72

B. 2) P(passes at least one of the tests) = P(pass test 1) + P(pass test 2) - P(pass both tests)

= 0.8 + 0.9 - 0.72

= 0.98