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 knowing whether event $$ A $$ has occurred gives no information about whether event $$ B $$ has occurred, and vice versa.

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 happening, you multiply their individual probabilities.

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

In summary, to find the probability of two independent events occurring together, simply multiply their individual probabilities.
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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