Question

The probability distribution of a discrete random variable $X$ is given as

Find the value of $P(2 < X < 6)$.

• A. $\frac{4}{21}$
• B. $\frac{1}{21}$
• C. $\frac{10}{21}$
• D. $\frac{4}{7}$

Hint:

To minimize arithmetic steps during an exam, keep your fractions written in terms of the common denominator $21$ until the final step. Summing the coefficients directly ($3 + 4 + 5 = 12$) before introducing the fraction constant saves time and prevents simple addition errors.

Answer 1

The Correct Option is D

Step 1: Understanding the Question:
We are given a discrete probability distribution for a random variable $X$ that takes integer values from 1 to 6 with probabilities scaled by a constant factor $K$. We need to find the total probability for the strict open interval $2 < X < 6$.

Step 2: Key Formula or Approach:
For a valid probability mass function, the sum of all individual probabilities across the entire sample space must equal exactly 1: $$\sum P(X_i) = 1$$ We use this total sum condition to determine the numerical value of the constant $K$. Then, we calculate the target probability by summing the discrete values that fall strictly within the given boundaries: $$P(2 < X < 6) = P(X=3) + P(X=4) + P(X=5)$$

Step 3: Detailed Explanation:
The probability distribution mapped from the structure is: Sum all the probability terms and set the total equal to 1: $$K + 2K + 3K + 4K + 5K + 6K = 1$$ $$21K = 1 \implies K = \frac{1}{21}$$ Now, compute the target probability for the range $2 < X < 6$. Note that since the inequalities are strict ($<$), the boundary values 2 and 6 are excluded from the sum: $$P(2 < X < 6) = P(X=3) + P(X=4) + P(X=5)$$ Substitute the expressions in terms of $K$: $$P(2 < X < 6) = 3K + 4K + 5K = 12K$$ Substitute the calculated value of $K = \frac{1}{21}$ into this expression: $$P(2 < X < 6) = 12 \times \frac{1}{21} = \frac{12}{21}$$ Reduce the fraction to its simplest terms by dividing both the numerator and the denominator by their greatest common divisor, 3: $$P(2 < X < 6) = \frac{4}{7}$$ This matches option (D).

Step 4: Final Answer:
The probability $P(2 < X < 6)$ is equal to $\frac{4}{7}$, which corresponds to option (D).