What indicates two events are independent in probability?

Two events are considered independent in probability when the occurrence of one event has no effect on the occurrence of the other. This means that knowing the outcome of one event does not provide any information about whether the other event will occur.

For example, if you roll a die and flip a coin, the result of the die roll does not influence the outcome of the coin flip. Thus, the events (rolling a specific number on the die and getting heads or tails on the coin) are independent.

Mutual exclusivity, which refers to situations where the events cannot happen at the same time, would indicate that if one event occurs, the other cannot occur. This is a different concept from independence since mutually exclusive events are inherently dependent; knowing that one event occurred means the other did not.

Events that cannot occur simultaneously would imply mutual exclusivity rather than independence. If two events are mutually exclusive, the occurrence of one directly prevents the occurrence of the other.

Having the same probability of occurrence does not relate to independence. Two events can have equal probabilities but still be dependent, as the occurrence of one could still affect the likelihood of the other occurring.

Understanding these distinctions clarifies why the indication of independence is based solely on the lack of influence between the outcomes