Every item made in a factory goes through three inspection stages, I, II and III, in this order. An item goes on to the next stage only if it passes the present stage; an item that fails at any stage is rejected. The probability that an item passes stage I is 0.9. If it has passed stage I, the probability that it passes stage II is 0.8. If it has passed stages I and II, the probability that it passes stage III is 0.5. Consider the following statements:
(A) The probability that an item passes stages I and II is 0.72.
(B) The probability that an item passes all three stages is 0.36.
(C) The probability that an item is rejected at stage II is 0.18.
(D) The probability that an item is rejected at stage III is 0.5.
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → (A), (B) and (C) only
Let \(E_1\), \(E_2\), \(E_3\) be the events that the item passes stages I, II and III. The given figures are
\(P(E_1) = 0.9,\quad P(E_2|E_1) = 0.8,\quad P(E_3|E_1\cap E_2) = 0.5\)
By the multiplication theorem:
\(P(E_1\cap E_2) = 0.9\times 0.8 = 0.72\)
\(P(E_1\cap E_2\cap E_3) = 0.72\times 0.5 = 0.36\)
Rejected at stage II means it passes stage I and then fails stage II; rejected at stage III means it passes stages I and II and then fails stage III:
\(P(E_1\cap E_2') = 0.9\times 0.2 = 0.18\)
\(P(E_1\cap E_2\cap E_3') = 0.72\times 0.5 = 0.36\)
So the item passes stages I and II with probability 0.72, passes all three with probability 0.36, is rejected at stage II with probability 0.18 and is rejected at stage III with probability 0.36.
Checking the statements
(A) The probability that an item passes stages I and II is 0.72. — Correct. \(0.9\times 0.8 = 0.72\).
(B) The probability that an item passes all three stages is 0.36. — Correct. \(0.9\times 0.8\times 0.5 = 0.36\).
(C) The probability that an item is rejected at stage II is 0.18. — Correct. \(0.9\times 0.2 = 0.18\).
(D) The probability that an item is rejected at stage III is 0.5. — Incorrect. The item must first pass stages I and II, so the probability is \(0.72\times 0.5 = 0.36\).