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. If Events A and B are Independent:** When two events are independent, the occurrence of one event does not affect the probability of the other. In this case, the probability of both events occurring together is the **product** of their individual probabilities. \[ P(A \text{ and } B) = P(A) \times P(B) \] Given: - \( P(A) = 0.5 \) - \( P(B) = 0.3 \) \[ P(A \text{ and } B) = 0.5 \times 0.3 = 0.15 \] **So, if A and B are independent, the probability that both occur is 0.15 (or 15%).** ### **2. If Events A and B are Dependent:** If the events are **dependent**, the occurrence of one event affects the probability of the other. In such cases, we need additional information, such as the **conditional probability** \( P(A|B) \) or \( P(B|A) \), to calculate \( P(A \text{ and } B) \). The formula for dependent events is: \[ P(A \text{ and } B) = P(A) \times P(B|A) = P(B) \times P(A|B) \] **Without information on the dependency between A and B (i.e., \( P(B|A) \) or \( P(A|B) \)), we cannot determine the exact probability.** ### **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 0.15 (15 %).**