E. Chapter 3 HW - Probability Topics Al Category: Homework (c) Are \( C \) and \( D \) independent events? Why or why not? A \( C \) and \( D \) are independent because \( \mathrm{P}(C \) AND \( D) \neq 0 \). B \( C \) and \( D \) are not independent because \( \mathrm{P}(C \mid D) \neq \mathrm{P}(C) \). C \( C \) and \( D \) are independent because \( \mathrm{P}(C \mid D) \neq \mathrm{P}(C)+\mathrm{P}(D) \). D Your answer is incorrect.

To determine if events \( C \) and \( D \) are independent, you need to check if the probability of both events occurring together equals the product of their individual probabilities: \( \mathrm{P}(C \cap D) = \mathrm{P}(C) \cdot \mathrm{P}(D) \). If this holds true, then they are independent; otherwise, they are dependent. In real-life situations, understanding independence is crucial. For example, consider a game of dice: rolling a six on one die does not influence the outcome of another die. Therefore, the events are independent. This principle helps in making decisions based on probabilities in activities like betting or in fields like statistics and genetics where independence can significantly alter outcomes and predictions!