3.3 Two basic rules of probability

When calculating probability, there are two rules to consider when determining if two events are independent or dependent and if they are mutually exclusive or not.

The multiplication rule

If A and B are two events defined on a sample space , then: $P(A\cap B)=P\left(B\right)P\left(A\right|B)$ .

This rule may also be written as: $P\left(A\right|B)=\frac{P(A\cap B)}{P(B)}$

(The probability of A given B equals the probability of A and B divided by the probability of B .)

If A and B are independent , then $P\left(A\right|B)=P(A)$ . Then $P(A\cap B)=P\left(A\right|B\left)P\right(B)$ becomes $P(A\cap B)=P\left(A\right)\left(B\right)$ .

One easy way to remember the multiplication rule is that the word "and" means that the event has to satisfy two conditions. For example the name drawn from the class roster is to be both a female and a sophomore. It is harder to satisfy two conditions than only one and of course when we multiply fractions the result is always smaller. This reflects the increasing difficulty satisfying two conditions.

The addition rule

If A and B are defined on a sample space, then: $P(A\cup B)=P\left(A\right)+P\left(B\right)-P(A\cap B)$ .

If A and B are mutually exclusive , then $P(A\cap B)=0$ . Then $P(A\cup B)=P\left(A\right)+P\left(B\right)-P(A\cap B)$ becomes $P(A\cup B)=P\left(A\right)+P\left(B\right)$ .