Question

Person A can solve 90% of the problems given in the book and Person B can solve 70%. Then the probability that atleast one of them will solve the problem selected at random from the book is

• A. 0.16
• B. 0.69
• C. 0.97
• D. 0.20

Hint:

P(at least one event occurs) = 1 - P(none of the events occur).
For independent events A and B: \(P(A \cup B) = P(A) + P(B) - P(A)P(B)\) \(P(A' \cap B') = P(A')P(B') = (1-P(A))(1-P(B))\)

Answer 1

The Correct Option is C

Let A be the event that Person A solves the problem. Let B be the event that Person B solves the problem. Given: \(P(A) = 90% = 0.90\) \(P(B) = 70% = 0.70\) We assume that the events A and B are independent. Probability that Person A does not solve the problem: \(P(A') = 1 - P(A) = 1 - 0.90 = 0.10\). Probability that Person B does not solve the problem: \(P(B') = 1 - P(B) = 1 - 0.70 = 0.30\). We need to find the probability that at least one of them will solve the problem. This is \(P(A \cup B)\). \(P(A \cup B) = P(A) + P(B) - P(A \cap B)\). Since A and B are independent, \(P(A \cap B) = P(A)P(B) = 0.90 \times 0.70 = 0.63\). So, \(P(A \cup B) = 0.90 + 0.70 - 0.63 = 1.60 - 0.63 = 0.97\). Alternatively, the probability that at least one solves the problem is 1 minus the probability that neither solves the problem. P(neither solves) = \(P(A' \cap B')\). Since A and B are independent, A' and B' are also independent. \(P(A' \cap B') = P(A')P(B') = 0.10 \times 0.30 = 0.03\). P(at least one solves) = \(1 - P(\text{neither solves}) = 1 - 0.03 = 0.97\). This matches option (c). \[ \boxed{0.97} \]