Step 1: Understanding the Concept:
We need to find the probability of $P(A' | B)$. Since events $A$ and $B$ are independent, their complements are also independent of each other.
The addition rule of probability is $P(A \cup B) = P(A) + P(B) - P(A \cap B)$. For independent events, $P(A \cap B) = P(A)P(B)$.
Step 2: Key Formula or Approach:
1. Use $P(A \cup B) = P(A) + P(B) - P(A)P(B)$ to find $P(A)$.
2. Calculate $P(A' | B)$. Since $A'$ and $B$ are independent, $P(A' | B) = P(A')$.
3. Substitute this value as $x$ in the given quadratic equations to see which one evaluates to 0.
Step 3: Detailed Explanation:
Given: $P(B) = \frac{2}{5}$, $P(A \cup B) = \frac{11}{20}$.
Let $P(A) = p$.
Using the formula for independent events:
\[ P(A \cup B) = P(A) + P(B) - P(A)P(B) \]
\[ \frac{11}{20} = p + \frac{2}{5} - p\left(\frac{2}{5}\right) \]
\[ \frac{11}{20} = \frac{2}{5} + p\left(1 - \frac{2}{5}\right) \]
\[ \frac{11}{20} - \frac{2}{5} = p\left(\frac{3}{5}\right) \]
Find common denominator (20) for the left side:
\[ \frac{11}{20} - \frac{8}{20} = \frac{3p}{5} \]
\[ \frac{3}{20} = \frac{3p}{5} \]
Cross-multiply to solve for $p$:
\[ 15 = 60p \implies p = \frac{15}{60} = \frac{1}{4} \]
So, $P(A) = \frac{1}{4}$.
We need to find $P(A' | B)$. By the definition of conditional probability:
\[ P(A' | B) = \frac{P(A' \cap B)}{P(B)} \]
Since $A$ and $B$ are independent, $A'$ and $B$ are also independent. Thus, $P(A' \cap B) = P(A')P(B)$.
\[ P(A' | B) = \frac{P(A')P(B)}{P(B)} = P(A') \]
Now find $P(A')$:
\[ P(A') = 1 - P(A) = 1 - \frac{1}{4} = \frac{3}{4} \]
The root is $x = \frac{3}{4}$. Now test this root in the given options.
Testing Option (A): $4x^2 - 7x + 3 = 0$
\[ 4\left(\frac{3}{4}\right)^2 - 7\left(\frac{3}{4}\right) + 3 = 4\left(\frac{9}{16}\right) - \frac{21}{4} + \frac{12}{4} = \frac{9}{4} - \frac{21}{4} + \frac{12}{4} = \frac{21 - 21}{4} = 0 \]
Since it equals 0, $x = 3/4$ is a root of this equation.
Step 4: Final Answer:
The correct equation is $4x^2 - 7x + 3 = 0$.