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 these events are **independent** or **dependent**. 1. **If Events \( A \) and \( B \) Are Independent:** When two events are independent, the occurrence of one does not affect the probability of the other occurring. In this case, the probability that both events \( A \) and \( B \) occur is the product of their individual probabilities. \[ P(A \text{ and } B) = P(A) \times P(B) = 0.5 \times 0.3 = 0.15 \] **Probability:** \( 0.15 \) 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, additional information about the relationship between \( A \) and \( B \) (such as conditional probabilities) is required to compute \( P(A \text{ and } B) \). Without this information, we cannot determine the exact probability. **Conclusion:** - **Assuming \( A \) and \( B \) are independent, the probability that both occur is \( 0.15 \) (or 15%).** - **If they are not independent, more information is needed to calculate the probability.** **Final Answer:** Assuming A and B are independent, the probability of both occurring is 0.5 × 0.3 = 0.15