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, denoted as $$ P(A \cap B) $$, we need to know whether the events are **independent** or **dependent**.

### 1. **If Events $$ A $$ and $$ B $$ Are Independent:**

Two events are **independent** if the occurrence of one does not affect the probability of the other. Under this assumption, the probability of both events occurring is the product of their individual probabilities:

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

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

### 2. **If Events $$ A $$ and $$ B $$ Are Not Independent:**

Without the independence assumption, we don't have enough information to determine $$ P(A \cap B) $$ precisely. However, we can establish bounds based on the principles of probability:

- **Maximum Possible Probability:** The probability of both events occurring cannot exceed the probability of the less likely event. Thus,

$$
P(A \cap B) \leq \min(P(A), P(B)) = \min(0.5, 0.3) = 0.3
$$

- **Minimum Possible Probability:** Assuming the events could overlap to the greatest extent possible (but are not necessarily exclusive), the lower bound is:

$$
P(A \cap B) \geq P(A) + P(B) - 1 = 0.5 + 0.3 - 1 = -0.2
$$

Since probabilities cannot be negative, the practical lower bound is:

$$
P(A \cap B) \geq 0
$$

### **Summary:**

- **If independent:** $$ P(A \cap B) = 0.15 $$
- **If not independent:** $$ 0 \leq P(A \cap B) \leq 0.3 $$

**Without additional information about the relationship between $$ A $$ and $$ B $$, the most specific answer assuming independence is $$ 0.15 $$.**

**Answer:** Assuming the events are independent, the probability of both occurring is 0.15.
Assuming events A and B are independent, the probability that both occur is 0.15.

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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