E Chapter 3 HW- Probability Topics AAl Category: Homework score: 0.6 or 3 points 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 \( D) \). \( \mathrm{P}(C \) AND \( D)=P(C)+P(D) \)

To solve for \( \mathrm{P}(C \text{ AND } D) \), we can use the formula for conditional probability: \[ \mathrm{P}(C \mid D) = \frac{\mathrm{P}(C \text{ AND } D)}{\mathrm{P}(D)} \] Rearranging this gives: \[ \mathrm{P}(C \text{ AND } D) = \mathrm{P}(C \mid D) \times \mathrm{P}(D) \] Substituting the known values: \[ \mathrm{P}(C \text{ AND } D) = 0.32 \times 0.7 = 0.224 \] So, \( \mathrm{P}(C \text{ AND } D) = 0.224 \). Now, let’s look at a quick history of probability theory: The concept of probability dates back to the 16th century when gambling became widely popular. However, it wasn’t until the 17th century that mathematicians like Blaise Pascal and Pierre de Fermat laid its foundations through discussions on games of chance. Their work sparked a whole new branch of mathematics that deals with analyzing random events—leading us to the formulas we use today! For real-world applications, consider how probability is fundamental in fields like finance, insurance, and even sports. Financial analysts use probability to assess risks and forecast market trends. In insurance, companies calculate premiums based on the likelihood of events happening. Even in sports analytics, teams employ probability to strategize and enhance performance based on the chances of certain plays succeeding!