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 know 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. - **Calculation:** If \( A \) and \( B \) are independent, the probability that both occur is the product of their individual probabilities. \[ P(A \text{ and } B) = P(A) \times P(B) = 0.5 \times 0.3 = 0.15 \] 2. **Dependent Events:** - **Definition:** Two events are dependent if the occurrence of one affects the probability of the other. - **Calculation:** In this case, additional information about how \( A \) and \( B \) are related is needed to determine \( P(A \text{ and } B) \). **Conclusion:** - **If \( A \) and \( B \) are independent**, the probability that both events occur is **0.15**. - **If they are not independent**, more information is required to calculate the probability. **Assuming independence, the probability that both events occur is 0.15.**