If event A has a probability of \( 0.5 \) and event B has a probability of \( 0.3 \), what is the probability that both events occur?

To determine the probability that **both** events \( A \) and \( B \) occur, we need to consider whether the events are **independent** or **dependent**. ### 1. **Independent Events** **Definition:** Two events are independent if the occurrence of one does not affect the probability of the other occurring. **Calculation:** If \( A \) and \( B \) are independent, the probability that both occur is the product of their individual probabilities: \[ P(A \text{ and } B) = P(A) \times P(B) \] **Given:** \[ P(A) = 0.5 \\ P(B) = 0.3 \] **So:** \[ P(A \text{ and } B) = 0.5 \times 0.3 = 0.15 \] **Therefore, if \( A \) and \( B \) are independent, the probability that both occur is **0.15** (or 15%). ### 2. **Dependent Events** If \( A \) and \( B \) are **dependent**, the occurrence of one affects the probability of the other. In this case, additional information is needed to determine the exact probability. Specifically, you would need to know either: - **Conditional Probability:** \( P(B|A) \) (the probability of \( B \) occurring given that \( A \) has occurred), or - **Joint Probability:** Directly provided \( P(A \text{ and } B) \). **Without this information**, we cannot accurately calculate \( P(A \text{ and } B) \) for dependent events. ### **Conclusion** - **If \( A \) and \( B \) are independent:** \( P(A \text{ and } B) = 0.15 \) (15%) - **If \( A \) and \( B \) are dependent:** Additional information is required to determine \( P(A \text{ and } B) \). --- **Assuming Independence**, the probability that both events occur is: \[ P(A \text{ and } B) = 0.5 \times 0.3 = 0.15 \quad \text{(15%)} \]