What does it mean for two events to be independent?

When two events are described as independent, it means that the occurrence of one event has no effect on the occurrence of the other. In other words, knowing that one event has occurred does not provide any information about the likelihood that the other event will occur.

For instance, consider the events of flipping a coin and rolling a die. The result of the coin flip (heads or tails) does not influence the result of the die roll (1 through 6). This is a clear illustration of independence, as the probability of rolling a certain number remains unchanged regardless of the coin flip's outcome.

This concept is fundamental in probability theory, distinguishing independent events from dependent events, where the outcome of one event clearly influences the outcome of another. Understanding independence is crucial in various applications, including statistics, risk assessment, and decision-making processes.