Absolutely, let's go through each part step by step. **Given:** \( X \) is a uniform random variable taking integer values \( 1, 2, \ldots, 13 \). This means each value from 1 to 13 is equally likely. There are **13** possible values. The probability for each value: \[ P(X = x) = \frac{1}{13} \] for \( x = 1, 2, \ldots, 13 \). --- #### a. \( P(X = 5) \) Since each value is equally likely: \[ P(X = 5) = \frac{1}{13} \] --- #### b. \( P(X5) \) It looks like you might have a typo here. Did you mean \( P(X < 5) \) or \( P(X \leq 5) \)? - If you meant \( P(X < 5) \), that's the probability that \( X \) is 1, 2, 3, or 4. - If you meant \( P(X \leq 5) \), that's the probability that \( X \) is 1, 2, 3, 4, or 5. I'll do both for you: - \( P(X < 5) = P(X = 1) + P(X = 2) + P(X = 3) + P(X = 4) = 4 \times \frac{1}{13} = \frac{4}{13} \) - \( P(X \leq 5) = P(X = 1) + ... + P(X = 5) = 5 \times \frac{1}{13} = \frac{5}{13} \) --- #### c. \( P(X > 3) \) This is the probability that \( X \) is 4, 5, ..., 13. How many values is that? From 4 to 13 (inclusive): \( 13 - 3 = 10 \) values. So, \[ P(X > 3) = \frac{10}{13} \] --- #### d. \( P(3X11) \) This notation is a bit unclear, but in probability, \( P(3 < X < 11) \) usually means the probability that \( X \) is strictly between 3 and 11, i.e., \( X = 4, 5, 6, 7, 8, 9, 10 \). How many values is that? From 4 to 10 inclusive: \( 10 - 4 + 1 = 7 \) values. So, \[ P(3 < X < 11) = \frac{7}{13} \] If you meant inclusive endpoints, i.e., \( P(3 \leq X \leq 11) \): From 3 to 11 inclusive: \( 11 - 3 + 1 = 9 \) values So, \( \frac{9}{13} \). Let me know if you meant something different! --- ### **Summary Table** | Probability | Answer | |-----------------------|------------| | a. \( P(X = 5) \) | \( \frac{1}{13} \) | | b. \( P(X < 5) \) | \( \frac{4}{13} \) | | c. \( P(X > 3) \) | \( \frac{10}{13} \) | | d. \( P(3 < X < 11) \)| \( \frac{7}{13} \) | --- **Let me know if you want clarification or if you meant different inequalities!**