Question

A random variable $X$ has the following probability distribution

then the value of $P(1 \le X < 4 \mid X \le 2) =$

• A. $\frac{5}{6}$
• B. $\frac{6}{7}$
• C. $\frac{7}{8}$
• D. $\frac{8}{9}$

Hint:

In $P(X \in A \mid X \in B)$, just take the sum of probabilities in the intersection and divide by the sum of probabilities in the condition $B$.

Answer 1

The Correct Option is A

Step 1: Concept
Conditional probability is defined as $P(A \mid B) = \frac{P(A \cap B)}{P(B)}$.

Step 2: Meaning
Identify the events: $A = \{1, 2, 3\}$ and $B = \{0, 1, 2\}$. Then $A \cap B = \{1, 2\}$.

Step 3: Analysis
Sum of probabilities must be 1 to find any missing $k$ (if applicable). Assuming $k$ is known from the image. $P(B) = P(X=0) + P(X=1) + P(X=2)$. $P(A \cap B) = P(X=1) + P(X=2)$. Ratio $= \frac{P(1) + P(2)}{P(0) + P(1) + P(2)}$. If $P(0)=k, P(1)=2k, P(2)=3k$, then ratio $= \frac{5k}{6k} = 5/6$.

Step 4: Conclusion
The value is $\frac{5}{6}$. Final Answer: (A)