Question

The range of a random variable \( X \) is \( \{0, 1, 2\} \). If \( P(X=0) = 3C^3 \), \( P(X=1) = 4C-10C^2 \), and \( P(X=2) = 5C-1 \), then the value of \( C \) is:

• A. \( \frac{2}{3} \)
• B. \( \frac{1}{3} \)
• C. \( \frac{5}{3} \)
• D. \( \frac{4}{3} \)

Hint:

When dealing with probabilities, remember that the sum of the probabilities of all possible outcomes must be equal to 1.

Answer 1

The Correct Option is B

Given that the range of the random variable \( X \) is \( \{0, 1, 2\} \), the total probability must add up to 1. That is, \[ P(X=0) + P(X=1) + P(X=2) = 1 \] Substitute the given probabilities into this equation: \[ 3C^3 + (4C - 10C^2) + (5C - 1) = 1 \] Simplify the equation: \[ 3C^3 - 10C^2 + 9C - 1 = 1 \] \[ 3C^3 - 10C^2 + 9C - 2 = 0 \] Solve for \( C \), which gives the value \( C = \frac{1}{3} \). Thus, the correct answer is \( \frac{1}{3} \).