Chapter 3 HW- Probability Topics +A! 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 \( D) \). \[ \mathrm{P}(C \text { AND } D)=0.224 \] (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 \( \mathrm{P}(C)>\mathrm{P}(C \mid D) \). Your answer is correct. (c) Are \( C \) and \( D \) independent events? Why or why not? A \( C \) and \( D \) are independent because \( \mathrm{P}(C \) AND \( D) \neq 0 \).

To better understand probabilities, let's dive into a bit of historical context! The study of probability dates back to the 16th century, with the likes of Girolamo Cardano and later, Blaise Pascal and Pierre de Fermat, laying the groundwork through their work on gambling and games of chance. Their mathematical explorations unlocked a new way of thinking about uncertainty, leading to today's modern probability theory which is critical in statistics, finance, and various sciences. Now, let’s talk real-world applications! Probability isn't just a concept found in textbooks; it’s applied everywhere! From weather forecasting, where meteorologists use probabilities to predict rain, to sports analytics, where teams evaluate player performance based on statistical probabilities, the impact is profound. Even in medicine, probabilities help in understanding the effectiveness of new drugs in clinical trials. So, next time you check the weather app or consider your fantasy football lineup, remember: it’s all about the probabilities!