6:34PM Thu Jan 23 In Emilio Mejia Felici... AA. 12 Independence and conditional probability If \( A \) and \( B \) are independent events, what is the probability that \( A \) and \( B \) both occur? Simplify any fractions. \( \square \)

When two events \( A \) and \( B \) are independent, the probability of both events occurring is calculated by multiplying their individual probabilities. So, if the probability of event \( A \) occurring is \( P(A) \) and the probability of event \( B \) occurring is \( P(B) \), then the probability of both events occurring is given by \( P(A \cap B) = P(A) \times P(B) \). For instance, if \( P(A) = \frac{1}{4} \) and \( P(B) = \frac{1}{3} \), the combined probability is \( P(A \cap B) = \frac{1}{4} \times \frac{1}{3} = \frac{1}{12} \). So, the answer you would put in the box is \( \frac{1}{12} \) after simplification. Understanding independent events can be quite fun! In real-world examples, consider flipping a coin and rolling a die. The coin landing heads (event \( A \)) is independent of the die rolling a four (event \( B \)). This illustrates how events can happen without influencing each other, making probability calculations simpler and more intuitive!