Question

Two students A and B appear for an exam. Probability that A passes is 0.05, B passes is 0.10, and both pass is 0.02. Find probability that neither passes.

• A. 0.87
• B. 0.98
• C. 1
• D. 0.11

Hint:

Whenever a question asks for the probability that none of the given events occur, first find the probability that at least one event occurs and then use \[ P(\text{None}) = 1-P(\text{At least one}). \]

Answer 1

The Correct Option is A

Concept:
This problem is based on the addition rule of probability and the complement rule. For any two events \(A\) and \(B\), \[ P(A\cup B)=P(A)+P(B)-P(A\cap B). \] The probability that neither event occurs is \[ P(\text{Neither}) = 1-P(A\cup B). \]

Step 1: Write the given probabilities. \[ P(A)=0.05, \qquad P(B)=0.10, \qquad P(A\cap B)=0.02. \]

Step 2: Find the probability that at least one student passes. Using \[ P(A\cup B)=P(A)+P(B)-P(A\cap B), \] we get \[ P(A\cup B) = 0.05+0.10-0.02. \] \[ P(A\cup B) = 0.13. \]

Step 3: Find the probability that neither student passes. \[ P(\text{Neither}) = 1-P(A\cup B). \] \[ P(\text{Neither}) = 1-0.13. \] \[ P(\text{Neither}) = 0.87. \] Conclusion: \[ \boxed{0.87} \] Hence, the probability that neither student passes is \(0.87\).