Theory question 115
Probability(a) A manufacturer produces light bulbs which are tested in the following way; A batch is accepted in either of the following cases: (i) a first sample of 5 bulbs contains no faulty bulbs; (ii) a first sample of 5 bulbs contains at least one faulty bulb but a second sample of size 5 contains no faulty bulb. If 10% of the bulbs are faulty, what is the probability that the batch is accepted?
(b) A bag contains 15 identical marbles of which 3 are black. Keshi picks a marble at random from the bag and replaces it. If this is repeated 10 times, what is the probability that he (i) did not pick a black ball? (ii) picked a black ball at most three times?
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(a) P(sample of 5 has no faulty bulb) = 0.95 = 0.5905
P(accepted) = 0.5905 + (1 − 0.5905)(0.5905) = 0.5905 + 0.2418 ≈ 0.832
(b) P(black) = 1/5, P(not black) = 4/5
(i) (4/5)10 ≈ 0.107
(ii) Σk=03 10Ck(1/5)k(4/5)10−k = 0.1074 + 0.2684 + 0.3020 + 0.2013 ≈ 0.879
Add the probabilities of the two separate ways a batch can be accepted.