Question

Student A's probability of solving the problem is \(1/2\), and B's is \(2/3\). What is the probability that the problem is solved?

• A. 4/6
• B. 1/3
• C. 5/6
• D. None of these

Hint:

Find the chance that both fail and subtract from 1, or use the union rule for independent events.

Answer 1

The Correct Option is C

Step 1: The problem is solved if at least one of the two students solves it. It is easiest to work with the complement, that is, the event that neither student solves it. Assume A and B work independently.

Step 2: Probability that A fails: \[P(\bar A) = 1 - \tfrac{1}{2} = \tfrac{1}{2}.\] Probability that B fails: \[P(\bar B) = 1 - \tfrac{2}{3} = \tfrac{1}{3}.\]

Step 3: Since the two events are independent, the probability that both fail is the product: \[P(\bar A \cap \bar B) = \tfrac{1}{2}\cdot \tfrac{1}{3} = \tfrac{1}{6}.\]

Step 4: The problem being solved is the complement of both failing: \[P(\text{solved}) = 1 - \tfrac{1}{6} = \tfrac{5}{6}.\]

Step 5: This matches option (C).

\[\boxed{\tfrac{5}{6}}\]