Until now we mostly used common sense and the (very versatile!) commands table() and prop.table().

But let's revisit the probability equations. We briefly summarize the probability-rules we need here, so that you don't have to look them up.

First an explanation of the notation:

• $P(A)$ = probability that event A occurs
• $P(A \cap B)$ = probability that events A and B occur together, the joint probability of A and B
• $P(A \cup B)$= probability that event A or event B occurs
• $P(A|B)$ = probability of A given that B occurs (= the probability of A conditional on B)

The four probability equations:

$$\begin{eqnarray*} P(A)& =& 1 − P(A^c)\\ P(A \cap B)& =& P(A|B)P(B) = P(B|A)P(A)\\ P(A \cup B)& =& P(A) + P(B) − P(A \cap B)\\ P(A|B)& =& \frac{P(B|A)P(A)}{P(B)}\\ \end{eqnarray*}$$

Note that the last equation (known as Bayes law) is in fact not really a new equation, but is just a rearrangement of the two alternative formulas for $P(A \cap B)$ at the right.