Question

The probability distribution of a random variable \( X \) is given as follows. Then, \( P(X = 50) - \frac{P(X \leq 30)}{P(X \geq 20)} \) = 

• A. \( \frac{2}{3} \)
• B. \( \frac{5}{6} \)
• C. \( \frac{1}{12} \)
• D. 0

Hint:

When dealing with probabilities, ensure that the total probability equals 1, and simplify fractions carefully when applying the probability formula.

Answer 1

The Correct Option is C

We are given the probability distribution for a random variable \( X \) with values at \( X = 10, 20, 30, 40, 50 \) and corresponding probabilities. 
The total sum of probabilities must be 1, so we start by solving for \( k \) using the equation: \[ k + 2k + 3k + 4k + 5k = 1 ⇒ 15k = 1 ⇒ k = \frac{1}{15}. \] Thus, the individual probabilities are: \[ P(X = 10) = \frac{1}{15}, P(X = 20) = \frac{2}{15}, P(X = 30) = \frac{3}{15}, P(X = 40) = \frac{4}{15}, P(X = 50) = \frac{5}{15}. \] Now, to find \( P(X = 50) - \frac{P(X \leq 30)}{P(X \geq 20)} \), we first calculate the cumulative probabilities: \[ P(X \leq 30) = P(X = 10) + P(X = 20) + P(X = 30) = \frac{1}{15} + \frac{2}{15} + \frac{3}{15} = \frac{6}{15} = \frac{2}{5}, \] \[ P(X \geq 20) = P(X = 20) + P(X = 30) + P(X = 40) + P(X = 50) = \frac{2}{15} + \frac{3}{15} + \frac{4}{15} + \frac{5}{15} = \frac{14}{15}. \] Substitute these values into the expression: \[ P(X = 50) - \frac{P(X \leq 30)}{P(X \geq 20)} = \frac{5}{15} - \frac{\frac{2}{5}}{\frac{14}{15}} = \frac{5}{15} - \frac{2}{5} \times \frac{15}{14} = \frac{5}{15} - \frac{6}{14} = \frac{1}{12}. \]