Marbles probability question

Question

6 marbles, 3 of which is defective.

Upon inspection on defective marble, 80% chance to successfully confirm indeed defective.

However, inspection made on good marble has a 5% chance of wrongly confirming defective.

3 marbles chosen at random from 6 available, with inspection done individually on each marble, find probability that test will deem all 3 marbles defective.

Please help to explain step by step, thank you.

Answer 1

Hi Nur, for the purpose of the problem, imagine the marbles are "labeled" as either D (defective) or G (good). There are 3 identical D marbles and 3 identical G marbles. As a result, there are a total of 4 distinct ways to draw 3 marbles out of the bag of 6 marbles:

3 D marbles --> the test will deem all 3 defective w/ prob = 0.8^3 = 0.512

2 D marbles, 1 G marbles --> the test will deem all 3 defective w/ prob 0.8^2 * 0.05 = 0.032

1 D marbles, 2 G marbles --> the test will deem all 3 defective w/ prob 0.8 * 0.05^2 = 0.002

3 G marbles --> the test will deem all 3 defective w/ prob 0.05^3 = 0.000125

Hence, the test will deem all 3 marbles randomly drawn out the bag of 6 marbles defective with probability:

0.512 + 0.032*3 + 0.002*3 + 0.000125 = 0.614125

The factor 3 for 2 D + 1 G and 1 D + 2 G comes from the fact that the test is done individually on each marble, so we would have the test sequence:

2 D + 1 G case:

D D G

D G D

G D D

1 D + 2 G case:

D G G

G D G

G G D

Hope my answer helps.