If there are two variables, and , we might want combined probabilities involving both and . There are three ways of combining random variables.

**Joint Probability:**- First, we can ask
for the
probability
that takes the value
*and*takes the value . This is denoted (also written or ) and is called the joint probability of and . This is simply the probability that both things occur: is and is . **Conditional Probability:**- Another quantity is the conditional probability that
takes the value
*given*that takes the value . This is denoted . This is the probability that takes the value assuming that definitely takes the value (no matter how unlikely it is that takes the value ).The relationship between the two is

the probability of and is the probability of given times the probability of . **Bayes Rule:**- From the formula relating joint probability with conditional
probability, Bayes Rule follows, since

so,

Note that if and are independent, , the joint probability is the product of the independent probabilities.**Example:**- An example may illustrate the difference between the joint probability
and the conditional probability. Suppose we pick an arbitrary British
adult. We wonder about two things: is this person rich and did they
attend public school? Thus, we have two variables:
- the person is rich,
- the person went to public school.

Now, consider the conditional probability, . That is the probability that a British adult is rich

*given*the fact that they went to public school, i.e. the fraction of public school alumni who are rich. I would assume that this is closer to 1, adults who went to public school are likely to be rich (I assume this is true).

**Probability of or :**- There is a third way of combining and that you should be aware
of. That is the probability that
takes the value
*or*takes the value . For this the expression is

Thus, if and are mutually exclusive, the probabilities simply add.