Let's break down the problem step by step: **Given:** \( X \) is a uniform random variable over the values \( x = 1, 2, ..., 13 \). So, \( X \) takes integer values from 1 to 13, each with equal probability. Total possible values: 13 Probability of any single value: \[ P(X = k) = \frac{1}{13},\quad \text{for } k = 1,2,...,13 \] --- ### a. \( P(X=5) \) As established above: \[ P(X=5) = \frac{1}{13} \] --- ### b. \( P(X5) \) I believe this is a typo or formatting error. If you meant **\( P(X < 5) \)**, then: \( X \) can take values \( 1, 2, 3, 4 \) (values less than 5), so: Number of favorable outcomes: 4 Total outcomes: 13 \[ P(X<5) = \frac{4}{13} \] If you meant something else, please clarify. --- ### c. \( P(X > 3) \) Possible values: \( 4, 5, ..., 13 \) How many values? \( 13 - 3 = 10 \) \[ P(X > 3) = \frac{10}{13} \] --- ### d. \( P(3X11) \) I believe you meant \( P(3 < X < 11) \) (i.e., \( X \) is strictly between 3 and 11). Possible values: \( 4, 5, ..., 10 \) How many values? \( 10 - 4 + 1 = 7 \) \[ P(3 < X < 11) = \frac{7}{13} \] --- ## **Summary Table** | Part | Probability Statement | Value | |------|------------------------------|--------------| | a | \( P(X = 5) \) | \( \frac{1}{13} \) | | b | \( P(X < 5) \) (assumed) | \( \frac{4}{13} \) | | c | \( P(X > 3) \) | \( \frac{10}{13} \)| | d | \( P(3 < X < 11) \) | \( \frac{7}{13} \) | If "b" and "d" mean something different, please clarify!