What does it mean for two events to be independent in probability?

When two events are independent in probability, it means that the occurrence of one event does not affect the occurrence of the other. For example, flipping a coin and rolling a die are independent events; the outcome of the coin toss won't influence whether the die shows a 4 or any other number. Mathematically, this is represented by the formula P(A and B) = P(A) * P(B), where P(A and B) is the probability that both events occur. Independence is particularly important in statistics and decision-making processes. For instance, when analyzing data, knowing that two factors are independent simplifies calculations and predictions. In practice, this concept is often applied in areas like genetics, where the inheritance of one trait doesn't affect the inheritance of another, or in financial investments, where the performance of one asset might be independent of another, helping investors balance their portfolios more effectively!