Suppose that \( X \sim U(2,7) \). Determine the value of: \( P(X>6 \mid X>3) \) Type your answer...

To determine \( P(X > 6 \mid X > 3) \), we can use the definition of conditional probability: \[ P(X > 6 \mid X > 3) = \frac{P(X > 6 \cap X > 3)}{P(X > 3)}. \] Since \( X \sim U(2, 7) \), the total range of \( X \) is from 2 to 7. 1. **Calculate \( P(X > 6) \)**: The probability that \( X \) is greater than 6 can be found by finding the length of the interval from 6 to 7, divided by the length of the total interval from 2 to 7. \[ P(X > 6) = \frac{7 - 6}{7 - 2} = \frac{1}{5}. \] 2. **Calculate \( P(X > 3) \)**: Here, we find the length of the interval from 3 to 7. \[ P(X > 3) = \frac{7 - 3}{7 - 2} = \frac{4}{5}. \] 3. **Find \( P(X > 6 \cap X > 3) \)**: Since \( X > 6 \) is a stronger condition than \( X > 3 \) (if \( X > 6 \), it automatically satisfies \( X > 3 \)), we have: \[ P(X > 6 \cap X > 3) = P(X > 6) = \frac{1}{5}. \] 4. **Now, substitute into the formula**: \[ P(X > 6 \mid X > 3) = \frac{P(X > 6)}{P(X > 3)} = \frac{\frac{1}{5}}{\frac{4}{5}} = \frac{1}{4}. \] So, the answer is \( P(X > 6 \mid X > 3) = \frac{1}{4} \).