<p>If A be any event in sample space then the maximum value of \(3P(A) + 4P(\overline{A})\) is:</p>
Step-by-Step Solution
Key Concept: Express the function in terms of P(A) and use the constraint that P(A) + P(complement A) = 1 to find the extremum.
<p>Let $P(A) = p$, then $P(\overline{A}) = 1 - p$ where $0 \leq p \leq 1$.</p><p>We need to maximize: $f(p) = 3p + 4(1-p) = 3p + 4 - 4p = 4 - p$</p><p>Since $f(p) = 4 - p$ is a decreasing function, the maximum occurs at $p = 0$.</p><p>Maximum value = $4 - 0 = 4$</p>
Correct Answer: a