Question

In order to ensure the quality of concrete material used in a reinforced concrete construction, concrete cylinders are collected at random from concrete mixes delivered to the construction site by a mixing plant. Past records of concrete from the same plant show that 85% of concrete mixes are good or are of satisfactory quality. To further ensure that the concrete delivered on site is of good quality, the engineer requires that one cylinder among those collected each day be tested (after 7 days of curing) for minimum compressive strength. The test method is not perfect: the probability that a good-quality concrete cylinder will pass the test is 0.9, or that a poor-quality cylinder can pass the test is 0.1.

Q: If a second cylinder is tested, and it also does pass the test, what is the probability that the mix of concrete is of good quality?

Answer should be: 0.9978

Answer 1

Probability that concrete mixes are good or are of satisfactory quality = P(G) = 0.85

Probability that concrete mixes are bad/poor quality P(B) = 1 - P(G) = 1 - 0.85 = 0.15

For 1st Cylinder,

P(Pass | G) = 0.9

P(Pass | B) = 0.1

By law of total probability, probability that the cylinder pass the test,

= P(Pass) = P(Pass | G) P(G) + P(Pass | B) P(B)

= 0.9 * 0.85 + 0.1 * 0.15

= 0.78

For 2nd Cylinder,

Given good quality, probability it is passed in both test = P(Pass2 | G) = 0.9 * 0.9 = 0.81

Given good quality, probability it is passed in both test = P(Pass2 | B) = 0.1 * 0.1 = 0.01

By law of total probability, probability that the cylinder pass the second test,

= P(Pass2) = P(Pass2 | G) P(G) + P(Pass2 | B) P(B)

= 0.81 * 0.85 + 0.01 * 0.15

= 0.69

If a second cylinder is tested, and it also does pass the test, the probability that the mix of concrete is of good quality

= P(G | Pass2)

= P(Pass2 | G) * P(G) / P(Pass2) (By Bayes theorem)

=  0.81 * 0.85 / 0.69

= 0.9978261