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

To calculate the probability of two independent events occurring together, you can use the multiplication rule of probability. Here’s a step-by-step explanation: 1. **Define 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 \( A \) occurring is the same whether or not event \( B \) occurs, and vice versa. 2. **Identify Individual Probabilities**: Determine the probabilities of each event occurring individually. Let: - \( P(A) \) be the probability of event \( A \) occurring. - \( P(B) \) be the probability of event \( B \) occurring. 3. **Apply the Multiplication Rule**: For independent events, the probability of both events occurring together (the joint probability) is calculated by multiplying their individual probabilities: \[ P(A \text{ and } B) = P(A) \times P(B) \] 4. **Example Calculation**: - Suppose \( P(A) = 0.5 \) (the probability of event \( A \) occurring) and \( P(B) = 0.3 \) (the probability of event \( B \) occurring). - To find the probability of both events occurring together, you would calculate: \[ P(A \text{ and } B) = P(A) \times P(B) = 0.5 \times 0.3 = 0.15 \] This means there is a 15% chance that both events \( A \) and \( B \) will occur together. In summary, the probability of two independent events occurring together is simply the product of their individual probabilities.