Question

For an event $E$, $P(E) + P(\bar{E}) = x$, then the value of $x^2 - 3$ is :

• A. -2
• B. 2
• C. 1
• D. -1

Hint:

Complementary events are mutually exclusive and exhaustive.
Their sum is always 1, regardless of how complex the event is.
Always substitute the standard values directly to simplify such algebraic expressions.

Answer 1

The Correct Option is A

Step 1: Understanding the Question:
The question gives us a relation for an event $E$, where the sum of the probability of occurrence of the event, $P(E)$, and the probability of non-occurrence of the event, $P(\bar{E})$, is equal to $x$. We need to find the value of the algebraic expression $x^2 - 3$.

Step 2: Key Formula or Approach:
According to the fundamental principles of probability theory:
The sum of the probability of an event happening and the probability of it not happening is always equal to 1.
\[ P(E) + P(\bar{E}) = 1 \]
This means that the value of $x$ is constant and equal to 1. We will substitute $x = 1$ into the given expression to compute the final value.

Step 3: Detailed Explanation:

• 1. We are given the relation:
\[ P(E) + P(\bar{E}) = x \]

• 2. Since $E$ and $\bar{E}$ are complementary events, their probabilities must sum to 1:
\[ P(E) + P(\bar{E}) = 1 \]

• 3. Comparing both equations, we get:
\[ x = 1 \]

• 4. Now, we need to evaluate the expression $x^2 - 3$. Substituting $x = 1$ into the expression:
\[ x^2 - 3 = (1)^2 - 3 \]

• 5. Simplify the arithmetic expression:
\[ x^2 - 3 = 1 - 3 = -2 \]


Step 4: Final Answer:
The value of $x^2 - 3$ is $-2$, which corresponds to option (A).