Independent events

Independent events are events that have no connection with each other. The simplest way of thinking about them is using conditional probabilities: if A and B are independent then

P(A|B) = P(A)

In other words, the probability of A given that B has happened is the same as the probability of A - so whether B happens or not makes no difference at all.

There is an equivalent formulation that is easy to derive:

P(A|B) = P(A) = P(A∩B)/P(B)

which rearranging gives

P(A∩B) = P(A) * P(B)

which is the familiar expression of independent variables that you use whenever working out probabilties of n coin tosses or die rolls.

If you are asked whether two events are independent, given that you know the probabilities, you start by saying

The events are independent if and only if:

P(A|B) = P(A) or P(A∩B) = P(A) * P(B)

Then substitute the probabilities that you know to see whether the condition is true. You only need to use one of the expressions, and you can choose which one for maximum convenience - depending on what probabilities you have already calculated.