Explain how the probability of two independent events occurring together is calculated.

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To calculate the probability of two independent events occurring together, we use the multiplication rule of probability. Here’s a step-by-step explanation:

1. **Understanding Independent Events**: 
   - Two events, $$ A $$ and $$ B $$, are considered independent if the occurrence of one event does not affect the occurrence of the other. This means that the probability of event $$ A $$ happening does not change whether event $$ B $$ happens or not.

2. **Defining Probabilities**:
   - Let $$ P(A) $$ be the probability of event $$ A $$ occurring.
   - Let $$ P(B) $$ be the probability of event $$ B $$ occurring.

3. **Calculating Joint Probability**:
   - The probability of both events $$ A $$ and $$ B $$ occurring together (denoted as $$ P(A \cap B) $$) is calculated using the formula:
     $$
     P(A \cap B) = P(A) \times P(B)
     $$
   - This formula states that to find the probability of both independent events occurring, you multiply their individual probabilities.

4. **Example**:
   - Suppose the probability of rolling a 3 on a fair six-sided die (event $$ A $$) is $$ P(A) = \frac{1}{6} $$.
   - The probability of flipping heads on a fair coin (event $$ B $$) is $$ P(B) = \frac{1}{2} $$.
   - To find the probability of both rolling a 3 and flipping heads, we calculate:
     $$
     P(A \cap B) = P(A) \times P(B) = \frac{1}{6} \times \frac{1}{2} = \frac{1}{12}
     $$

5. **Conclusion**:
   - The joint probability of two independent events is simply the product of their individual probabilities. This principle is fundamental in probability theory and is widely used in various applications, including statistics, risk assessment, and decision-making processes.
To find the probability of two independent events both happening, multiply their individual probabilities.

Explain how the probability of two independent events occurring together is calculated.

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