Review of basic probability theory

We hope that the reader has seen a little basic probability
theory previously. We will give a very quick review; some
references for further reading appear at the end of the
chapter. A variable represents an event (a subset of the
space of possible outcomes). Equivalently, we can represent
the subset via a *random
variable* , which is a function from
outcomes to real numbers; the subset is the domain over which
the random variable has a particular
value.
Often we
will not know with certainty whether an event is true in the
world. We can ask the probability of the event
. For two events and , the
joint event of both events occurring is described by the joint probability
. The conditional probability expresses
the probability of event given that event occurred.
The fundamental relationship between joint and
conditional probabilities is given by the *chain
rule* :

Writing
for the complement of an event, we similarly have:

(57) |

(58) |

From these we can derive *Bayes' Rule* for inverting conditional probabilities:

Finally, it is often useful to talk about the *odds* of an event, which provide a kind of multiplier for how probabilities change:

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2009-04-07