What are independent events?

independant events are where the probability of one event does that influence the probability of the other happening.

If events A and B are independent, this means that the probability of one of the events happening does not affect the probability of the other event happening. This can be written as P(A|B) = P(A). This expression means the probability of event A happening given event B has happened (left-hand side) equals the probability that event A happens (right-hand side). Of course, if events A and B are independent, whether event B happens or not, should not have an effect on the probability that event A happens.

To express this formally: If events A and B are independent, then P(A|B) = P(A), similarly, also P(B|A) = P(B), and P(A and B) = P(A) P(B). The last condition "P(A and B) = P(A) P(B)", is derived from the other two expressions. Note that if one of these expressions is true, you can be certain that the other expressions are true as they are completely equivalent, and that the events A and B are independent. Note also sometimes it will be easier to check P(A and B) = P(A) P(B) than P(A|B) = P(A), and you should choose which expression to verify depending on the information you have and how easy it is to verify.

Independent events are events whose outcomes cannot affect one another. A more mathematical definition would be that events A and B are independent if and only if P(A = x) = P(A = x | B = y) for all x, y, and P(B = y) = P(B = y | A = x) for all x, y. In words, this means that the probability of A being equal to x is the same as the probability of A being equal to x given that B is equal to y, for all possible outcomes x and y, and vice versa. For example, if I toss a coin and roll a die, the probability of the coin toss being any particular outcome is 1/2, and the probability of the die being any particular outcome is 1/6. No matter what result I get for the coin toss, it won't affect the result I get for the die roll, and vice versa.

An event that does not depend on anything