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 these two events are **independent** or **dependent**.

### 1. **Independent Events**

If **events $$ A $$ and $$ B $$ are independent**, the occurrence of one event does not affect the probability of the other. In this case, the probability that both events occur together (denoted as $$ P(A \cap B) $$) is the product of their individual probabilities:

$$
P(A \cap B) = P(A) \times P(B)
$$

Given:
- $$ P(A) = 0.5 $$
- $$ P(B) = 0.3 $$

Calculating:

$$
P(A \cap B) = 0.5 \times 0.3 = 0.15
$$

**So, if $$ A $$ and $$ B $$ are independent, the probability that both occur is $$ 0.15 $$ or $$ 15\% $$.**

### 2. **Dependent Events**

If **events $$ A $$ and $$ B $$ are dependent**, the occurrence of one event affects the probability of the other. In this case, you need additional information about how $$ A $$ and $$ B $$ are related to determine $$ P(A \cap B) $$. Specifically, you would need to know either:
- $$ P(B | A) $$: The probability of $$ B $$ given that $$ A $$ has occurred.
  
  Then,
  $$
  P(A \cap B) = P(A) \times P(B | A)
  $$

- Or $$ P(A | B) $$: The probability of $$ A $$ given that $$ B $$ has occurred.

Then,
  $$
  P(A \cap B) = P(B) \times P(A | B)
  $$

**Without additional information on the dependence between $$ A $$ and $$ B $$, we cannot determine $$ P(A \cap B) $$ for dependent events.**

### **Conclusion**

- **If $$ A $$ and $$ B $$ are independent:**  
  $$ P(A \cap B) = 0.15 $$ or $$ 15\% $$.

- **If $$ A $$ and $$ B $$ are dependent:**  
  Additional information is required to determine $$ P(A \cap B) $$.

---

**Please let me know if you have information about the relationship between $$ A $$ and $$ B $$, or if you need further assistance!**
If events A and B are independent, the probability that both occur is 0.15 or 15%. If they are dependent, more information is needed to determine the probability.

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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