Let $A,B$ be independent events with $P(A) = P(B)$ and $P(A \cup B) = \frac{1}{2}$, then $P(A)$ is e

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Let $A,B$ be independent events with $P(A) = P(B)$ and $P(A \cup B) = \frac{1}{2}$, then $P(A)$ is equal to

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Sol. [Key Concept: $A,B$ independent, $P(A) = P(B)$, given $P(A \cup \ B)$: solve for $P(A)$.]$$P(A \cup B) = P(A) + P(B) - P(A \cap B)$$$$\begin{matrix} \Rightarrow & \frac{1}{2} = 2P(	ext{ }A) - (P(	ext{ }A))^{2} \ \Rightarrow & P(	ext{ }A) = 1 - \frac{1}{\sqrt{2}} \end{matrix}$$

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