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?

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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:**
Independent events mean that the occurrence of one event does not affect the probability of the other event occurring. In this case, the probability that both events 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:**
Dependent events mean that the occurrence of one event affects the probability of the other event occurring. Without additional information about how the events are related, we cannot determine the exact probability in this scenario.

**Example:**
- If we knew $$ P(B|A) $$ (the probability of B occurring given that A has occurred), then:
$$
P(A \text{ and } B) = P(A) \times P(B|A)
$$
But since we don't have this information, we can't calculate it.

### **Conclusion:**
Assuming that events **A** and **B** are **independent**, the probability that both events occur is **0.15** or **15%**.

**Answer:**  
Assuming the events are independent, the probability both occur is 0.5 × 0.3 = 0.15.
If events A and B are independent, the probability that both occur is 0.15 or 15%.

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?

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