1. **Problem 50:** Determine which given set of probabilities cannot be a discrete probability distribution. 2. **Recall the rules for a discrete probability distribution:** - Each probability $P(x)$ must satisfy $0 \leq P(x) \leq 1$. - The sum of all probabilities must equal 1, i.e., $\sum P(x) = 1$. 3. **Check each option:** - A: $0.10 + 0.48 + 0.39 + 0.13 = 1.10$ (sum is greater than 1, invalid) - B: $0.10 + 0.45 + 0.55 + 0.13 = 1.23$ (sum is greater than 1, invalid) - C: $0.10 + 0.45 + 0.35 = 0.90$ (sum less than 1 and missing probability for $x=3$, invalid) - D: $0.10 + 0.45 + 0.35 + 0.13 = 1.03$ (sum slightly greater than 1, invalid) 4. **Conclusion for Problem 50:** None of the options sum exactly to 1, so all violate the sum rule. However, option C is missing a probability for $x=3$, so it cannot be a valid discrete distribution as it is incomplete. --- 5. **Problem 51:** Find the missing probability for $x=1$ given the histogram bars: - $P(0) = 0.10$ - $P(2) \approx 0.3$ - $P(3) \approx 0.2$ - $P(1) = ?$ 6. **Use the sum rule:** $$P(0) + P(1) + P(2) + P(3) = 1$$ 7. Substitute known values: $$0.10 + P(1) + 0.3 + 0.2 = 1$$ 8. Simplify: $$P(1) + 0.60 = 1$$ 9. Solve for $P(1)$: $$P(1) = 1 - 0.60 = 0.40$$ 10. **Check options:** None of the options (0.5, 0.8, 0.7, 0.8) match 0.40 exactly, but the closest reasonable answer is 0.5 (option A). **Final answers:** - Problem 50: Option C cannot be a discrete probability distribution. - Problem 51: The missing probability is approximately 0.40, closest to option A (0.5).