CONCEPT 1 – The Addition Rule
As discussed in earlier objectives, we may want to calculate the probability of two or more options being a successful event. For example, in a bag of marbles there are 5 red, 2 green and 3 purple, maybe the successful event is either picking a red or a green. This language naturally leads us to want to add the two probabilities together. We succeed with either red or with green so we add them and get the probability of 7/10.
Be careful!!! While this seems to be the general formula for addition -- IT IS NOT!!! This works nicely here because these two events are MUTUALLY EXCLUSIVE. Let us look at another example and you will see how this rule will not work and need a bit of revision.
A single 12 sided die is rolled. Event A is the factors of 6 and event B is that the roll was even.
Let us look at why our addition formula worked for the first case and not the second.
|This is the critical concept... that the intersection gets double counted when adding two non mutually exclusive events. This set of three Venn diagrams is very important to display.
We had the same issue when working with a two way table. We had to be careful about not double counting the intersection of the two probabilities.
To take into account the possibility of double counting the intersection we need to modify the addition rule:
P(A or B) = P(A) + P(B) – P(A and B)
This formula works for ALL CASES. If the two events are have an intersection, then its duplication will be canceled out and if the two events are mutually exclusive the intersection is empty and it simplifies to adding the two probabilities together and subtracting nothing.
Here are a few examples: