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

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.