Question

If A and B are independent events such that \( P(A \cap B') = \frac{3}{25} \) and \( P(A' \cap B) = \frac{8}{25} \), then \( P(A) = \)}

• A. \( \frac{3}{8} \)
• B. 4
• C. \( \frac{1}{5} \)
• D. \( \frac{2}{5} \)

Hint:

For independent events, $P(A \cap B') = P(A)P(B')$. Use algebraic substitution to solve.

Answer 1

The Correct Option is C

Step 1: Independence Equations
$P(A)(1-P(B)) = 3/25$
$(1-P(A))P(B) = 8/25$
Step 2: Let \(P(A)=x, P(B)=y\)
$x(1-y) = 3/25 \implies x - xy = 3/25$
$(1-x)y = 8/25 \implies y - xy = 8/25$
Step 3: Subtract Equations
$y - x = 5/25 = 1/5 \implies y = x + 1/5$.
Substitute into first: $x(1 - x - 1/5) = 3/25 \implies x(4/5 - x) = 3/25$.
$25x(4/5 - x) = 3 \implies 20x - 25x^2 = 3$.
Step 4: Solve Quadratic
$25x^2 - 20x + 3 = 0 \implies (5x-3)(5x-1) = 0$.
$x = 1/5$ or $x = 3/5$.
Final Answer:(C)