Question

Let $A$ and $B$ are independent events with $\text{P(B)} = \frac{2}{5}, \text{P(A} \cup \text{B)} = \frac{11}{20}$, then $\text{P(A}' | \text{B)}$ is root of the equation}

• A. $4x^2 - 7x + 3 = 0$
• B. $4x^2 + 7x + 3 = 0$
• C. $4x^2 - 3x - 7 = 0$
• D. $6x^2 - 5x + 1 = 0$

Hint:

If A and B are independent, knowing B occurred does not change the probability of A.

Answer 1

The Correct Option is A

Step 1: Concept
For independent events, $P(A \cup B) = P(A) + P(B) - P(A)P(B)$ and $P(A' | B) = P(A')$.

Step 2: Meaning
$11/20 = P(A) + 2/5 - (2/5)P(A)$.
$11/20 - 8/20 = P(A)(3/5) \implies 3/20 = P(A)(3/5) \implies P(A) = 1/4$.

Step 3: Analysis
$P(A') = 1 - 1/4 = 3/4$. Since events are independent, $P(A' | B) = P(A') = 3/4$.

Step 4: Conclusion
Substitute $x = 3/4$ into equations: $4(9/16) - 7(3/4) + 3 = 9/4 - 21/4 + 12/4 = 0$. Thus, it's a root of equation (A). Final Answer: (A)