Question

Let $E_{1}$ and $E_{2}$ be two independent events. If $P(E_{1}' \cap E_{2}) = \frac{2}{15}$ and $P(E_{1} \cap E_{2}') = \frac{1}{6}$, then $P(E_{1})$ is ________.

• A. $\frac{2}{15}$
• B. $\frac{13}{15}$
• C. $\frac{2}{13}$
• D. $\frac{1}{5}$

Hint:

For independent events, their complements are also independent.

Answer 1

The Correct Option is D

Step 1: Concept
For independent events, $P(A \cap B) = P(A) \times P(B)$.
Step 2: Analysis
Let $P(E_1) = x$ and $P(E_2) = y$.
1. $P(E_1' \cap E_2) = (1-x)y = \frac{2}{15}$
2. $P(E_1 \cap E_2') = x(1-y) = \frac{1}{6}$
Step 3: Calculation
From (1): $y - xy = \frac{2}{15}$
From (2): $x - xy = \frac{1}{6}$
Subtracting the two equations: $x - y = \frac{1}{6} - \frac{2}{15} = \frac{5 - 4}{30} = \frac{1}{30}$.
$y = x - \frac{1}{30}$.
Substitute $y$ in (2): $x - x(x - \frac{1}{30}) = \frac{1}{6} \Rightarrow x - x^2 + \frac{x}{30} = \frac{1}{6}$.
Multiplying by 30: $30x - 30x^2 + x = 5 \Rightarrow 30x^2 - 31x + 5 = 0$.
Solving the quadratic: $(5x - 1)(6x - 5) = 0$.
$x = 1/5$ or $x = 5/6$. Comparing with options .
Step 4: Conclusion
Hence, $P(E_1) = 1/5$.
Final Answer:(D)