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Independent events in probability are defined as events where the occurrence of one event does not influence the occurrence of another event. This means that knowing the outcome of one event provides no information about the probability of the other event occurring.
For example, consider tossing a fair coin and rolling a die. The result of the coin toss (either heads or tails) does not affect the result of the die roll (which can result in any number from 1 to 6). Mathematically, if two events A and B are independent, the probability of both events occurring together can be represented as P(A and B) = P(A) × P(B).
In contrast, the other options describe different types of relationships between events. Some events may be mutually exclusive (cannot occur at the same time), while others might be dependent (where the outcome of one event affects the outcome of the other). Understanding the characteristics of independent events is crucial for accurately calculating probabilities in various scenarios in mathematics and statistics.