Probability
Independent Events
IIT-JEE 2011
Grade 12
Question:
Let $E$ and $F$ be two independent events. The probability that exactly one of them occurs is $\frac{11}{25}$ and the probability of none of them occurring is $\frac{2}{25}$. If $P(E) < P(F)$, then $P(F)$ equals:
(1) $\frac{3}{5}$
(2) $\frac{4}{5}$
(3) $\frac{2}{5}$
(4) $\frac{1}{5}$
Step-by-Step Solution
Key Concept: Translate the two given conditions into the equations $P(E) + P(F) - 2P(E)P(F) = \frac{11}{25}$ and $(1 - P(E))(1 - P(F)) = \frac{2}{25}$, then solve this symmetric pair for $P(E) + P(F)$ and $P(E) P(F)$.
The detailed step-by-step mathematical proof is available inside the Mathbee app workspace.
Correct Answer: (2)