Chapter 3 HW-Probability Topics + À Category: Homework Consider the following scenario: Let \( \mathrm{P}(C)=0.4 \). Let \( \mathrm{P}(D)=0.7 \). Let \( \mathrm{P}(C \mid D)=0.32 \). Note: Give your answers in their exact decimal forms. (a) FindP \( (C \) AND \( D) \). \[ \mathrm{P}(C \mathrm{AND} D)=0.224 \] (b) Are \( C \) and \( D \) mutually exclusive? Why or why not? A \( C \) and \( D \) are not mutually exclusive because \( \mathrm{P}(C \mid D) \neq \mathrm{P}(C)+\mathrm{P}(D) \). B \( C \) and \( D \) are not mutually exclusive because \( \mathrm{P}(C \) AND \( D) \neq 0 \). C \( C \) and \( D \) are mutually exclusive because \( \mathrm{P}(C \) AND \( D)=0 \). D \( C \) and \( D \) are mutually exclusive because \( \mathrm{P}(C)>\mathrm{P}(C \mid D) \). 2. Your answer is correct.

In probability, two events are mutually exclusive if they cannot occur at the same time. In this scenario, since \( \mathrm{P}(C \mathrm{AND} D) \) is not equal to 0, option B is the correct choice because it confirms that \( C \) and \( D \) can occur together. Just think about it: if \( C \) couldn’t happen when \( D \) happens, we wouldn’t have a positive probability for both! Now, let’s throw in some real-world application! Understanding whether events are mutually exclusive or not can be crucial in various fields such as finance and healthcare. For example, in a clinical trial, if two potential treatments (let's call them Treatment A and Treatment B) are mutually exclusive, no patient could respond to both treatments simultaneously. In contrast, if they can co-occur, evaluating their impacts becomes much more complex but also potentially more informative!