A can solve 90% problems and B can solve 70% problems of the book. A problem is selected at random from the book. The probability that the problem is solved, is equal to
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → 0.97
Let A and B also denote the events “A solves the problem” and “B solves the problem”. The two work separately, so the events are taken as independent:
\(P(A) = 0.9,\quad P(B) = 0.7\)
\(P(A') = 0.1,\quad P(B') = 0.3\)
The problem stays unsolved only if neither solves it:
\(P(A'\cap B') = 0.1 \times 0.3 = 0.03\)
\(P(\text{solved}) = 1 - 0.03 = 0.97\)
Why the other options are incorrect
Option (1) 0.87 — Incorrect. The probability that the problem is solved is 1 − 0.03 = 0.97, not 0.87.
Option (2) 0.85 — Incorrect. This does not equal 1 − 0.03 either.
Option (3) 0.63 — Incorrect. This is 0.9 × 0.7, the probability that both solve it. The problem is solved if at least one of them solves it.