E- Chapter 3 HW- Probability Topics 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.

To find \( \mathrm{P}(C \cap D) \), we can use the formula for conditional probability: \[ \mathrm{P}(C \cap D) = \mathrm{P}(C \mid D) \times \mathrm{P}(D) \] Substituting the given values: \[ \mathrm{P}(C \cap D) = 0.32 \times 0.7 = 0.224 \] Now, to find \( \mathrm{P}(D \mid C) \), we can use Bayes' theorem: \[ \mathrm{P}(D \mid C) = \frac{\mathrm{P}(C \cap D)}{\mathrm{P}(C)} \] Substituting the values we have: \[ \mathrm{P}(D \mid C) = \frac{0.224}{0.4} = 0.56 \] In summary, the probabilities you are looking for are: - \( \mathrm{P}(C \cap D) = 0.224 \) - \( \mathrm{P}(D \mid C) = 0.56 \)