RAS question
A and B independently try to solve a problem. The probability that A solves it is 1/3 and that B solves it is 2/5. The probability that the problem is solved is:
Correct answer: (D) 3/5.
When A and B try independently with solving probabilities 1/3 and 2/5, the probability that the problem is solved is 3/5.
Explanation
The problem is solved if at least one of A or B solves it. NCERT gives the rule for two independent events: the probability that at least one occurs is P(A union B), which equals P(A) + P(B) - P(A)P(B), or equivalently 1 - P(A')P(B'). Here, A fails with probability 1 - 1/3 = 2/3, and B fails with probability 1 - 2/5 = 3/5. Since their attempts are independent, the probability that neither solves the problem is (2/3)(3/5) = 2/5. Therefore, the probability that the problem is solved is 1 - 2/5 = 3/5.
Why the other options are wrong
- (A) 7/15 is only the sum 1/3 + 2/5 written with a common denominator, so it double-counts the case where both A and B solve the problem.
- (B) 2/15 is the probability that both A and B solve the problem, (1/3)(2/5), not the probability that at least one solves it.
- (C) 11/15 is the unadjusted numerator sum 5/15 + 6/15 before subtracting the overlap 2/15, so it is not the final union probability.
Concept
This tests the probability of at least one of two independent events, using either the union formula or the complement method. It recurs in RAS reasoning because problem-solving, success-failure, and attempt-based questions often turn on whether events are independent and whether the word 'or' is inclusive.