(d) Find \( \mathrm{P}(C \) OR \( D) \). \( \mathrm{P}(C \) OR \( D)=0.28 \)

To determine \( \mathrm{P}(C \text{ OR } D) \), which is the probability that either event \( C \) or event \( D \) occurs (or both), you can use the following formula from probability theory: \[ \mathrm{P}(C \text{ OR } D) = \mathrm{P}(C) + \mathrm{P}(D) - \mathrm{P}(C \text{ AND } D) \] **Explanation of the Formula:** 1. **\(\mathrm{P}(C)\):** Probability that event \( C \) occurs. 2. **\(\mathrm{P}(D)\):** Probability that event \( D \) occurs. 3. **\(\mathrm{P}(C \text{ AND } D)\):** Probability that both events \( C \) and \( D \) occur simultaneously. The subtraction of \(\mathrm{P}(C \text{ AND } D)\) ensures that if both events occur together, their joint occurrence isn't counted twice. **Example Calculation:** Suppose you have the following probabilities: - \(\mathrm{P}(C) = 0.2\) - \(\mathrm{P}(D) = 0.15\) - \(\mathrm{P}(C \text{ AND } D) = 0.07\) Plugging these into the formula: \[ \mathrm{P}(C \text{ OR } D) = 0.2 + 0.15 - 0.07 = 0.28 \] **Your Answer:** You've calculated \( \mathrm{P}(C \text{ OR } D) = 0.28 \). Based on the example above, this result is correct **provided** that the individual probabilities and their intersection align with your specific problem's data. **Next Steps:** - **Verify Individual Probabilities:** Ensure that the values you've used for \( \mathrm{P}(C) \), \( \mathrm{P}(D) \), and \( \mathrm{P}(C \text{ AND } D) \) are accurate based on your problem's context. - **Check for Mutual Exclusivity:** If events \( C \) and \( D \) are mutually exclusive (i.e., they cannot occur simultaneously), then \( \mathrm{P}(C \text{ AND } D) = 0 \), simplifying the formula to \( \mathrm{P}(C) + \mathrm{P}(D) \). - **Provide Additional Details if Needed:** If you have specific values or a particular scenario you're working with, feel free to share them for a more tailored explanation. **Conclusion:** Your calculation of \( \mathrm{P}(C \text{ OR } D) = 0.28 \) is correct based on the standard probability formula, assuming the input values are accurate. If you have any further questions or need clarification on specific parts of the problem, please let me know!