Independent Events are two events whose outcomes are not influenced by one another. By definition, we know that two events are independent if the probability of one event happening is the same as the probability of it happening given a second event. Your AP formula sheet writes it like this:
P(A) = P(A | B)
In plain english, this means that we have two events, A and B. On one side of the equation, we calculate the probability of event A happening. On the other side of the equation, we calculate the conditional probability of event A, given that event B has already happened. If they are the same on both sides of the equation, then we have independent events. If not, then the events are dependent on one another.
In context, "independent events" means that whatever event B is does not influence or change the probability of event A when it happens. For example, if a student of mine skips class to go watch a fight, that's not going to change the probability that I'm still going to be teaching class that period. So the student going to the fight and me teaching class would be independent events.
"Dependent events" means that event B somehow influences the outcomes of event A. Therefore, the probability of them happening together is different than if event A happened by itself. For example, the probability of me showing up to teach on a given day is probability about 98%. However, the probability of me showing up to teach given that I know I have a cold is significantly lower, probability around 30%. So since p(teaching) does not equal P(teaching given cold), those two events would be dependent on one another.
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