Dependent events

In probability and statistics, a dependent event has an outcome that is affected by the outcome of a previous event. This is in contrast to an independent event in which the outcome is not affected by a previous event.


There are three marbles in a bag. One of the marbles is blue, another green, and the third is red. One marble is picked out of the bag at random - a blue marble. The blue marble is not returned to the bag. What color marbles could be picked out of the bag the next time?

Since there is only one blue marble and it was picked out of the bag, the next marble picked cannot be blue. It can only be green or red. This is an example of a dependent event. The first event, removing the blue marble from the bag, affects the potential outcome of the second event.

If the blue marble were chosen, but then returned to the bag, this would instead be an example of an independent event, since the blue marble could be chosen the second time around with the same probability (⅓) as any of the other colored marbles being picked

Probability of dependent events

Since the outcome of one event affects the outcome of another event when working with dependent events, the probability of later events changes based on previous events. In the example above with the marbles, there is initially a ⅓ chance of any color of marble being chosen. However, after the blue marble is removed, there are only a red and green marble left. There is no chance that a blue marble can be picked, and a ½ chance of either a red or green marble being picked.

The probability of an event occuring given that another event has already occured is referred to as the conditional probability of the event occuring. Using probability notation, it is written as:


which is read as the conditional probability of event A given event B.

Again referencing the above example, if picking out the blue marble is event B, and picking out a red marble is event A, P(A|B) = 0.5, since there are only 2 marbles to choose from after the blue one has been removed, one of which is red.