Question

Let \( E_1, E_2 \) and \( E_3 \) be mutually independent events. Statement I: \( E_1 \) and \( E_2 \cup E_3 \) are independent. Statement II: \( E_1 \) and \( E_2 \cap E_3 \) are independent. Which one of the following options is correct?

• A. Both I and II are true
• B. Only I is true
• C. Only II is true
• D. Both I and II are false

Hint:

For mutually independent events: \[ P(A\cap B)=P(A)P(B) \] and \[ P(A\cap B\cap C)=P(A)P(B)P(C) \] These properties can often be extended to unions and intersections through algebraic manipulation of probabilities.

Answer 1

The Correct Option is A

Step 1: State what independence means.
Mutually independent events satisfy $P(E_i\cap E_j)=P(E_i)P(E_j)$ and $P(E_1\cap E_2\cap E_3)=P(E_1)P(E_2)P(E_3)$.
Step 2: Test Statement I.
Find $P(E_1\cap(E_2\cup E_3))=P((E_1\cap E_2)\cup(E_1\cap E_3))$. By the addition rule this is $P(E_1\cap E_2)+P(E_1\cap E_3)-P(E_1\cap E_2\cap E_3)$.
Step 3: Substitute independence.
This equals $P(E_1)[P(E_2)+P(E_3)-P(E_2)P(E_3)]=P(E_1)P(E_2\cup E_3)$. So $E_1$ and $E_2\cup E_3$ are independent; Statement I is true.
Step 4: Test Statement II.
$P(E_1\cap(E_2\cap E_3))=P(E_1\cap E_2\cap E_3)=P(E_1)P(E_2)P(E_3)$.
Step 5: Compare with the product.
Also $P(E_2\cap E_3)=P(E_2)P(E_3)$, so $P(E_1)P(E_2\cap E_3)=P(E_1)P(E_2)P(E_3)$, which matches. So Statement II is true.
Step 6: Conclude.
Both statements are true. \[ \boxed{\text{Both I and II are true}} \]