Indepdendence, while not a new concept in this class (it is
directly based on conditional probability), is another central concept
of probability theory. In terms of conditional probability, two events
$E,F\subset S$ of a probability space $S$ are called
independent if
$$
P(E|F)=P(E)\text{ or }P(F|E)=P(F)\text{ or }P(E\cap F)=P(E)P(F),
$$
which simply means that knowledge of the occurrence of one event does
not influence the probability of the other. That the definition is
symmetric is most obvious in the third characterization, which is a
consequence of the definition of conditional probability. As usual
various examples are presented.
Lecture 11
This lecture deals with examples of the use of conditional
independence and conditioning in the computation of more involved
problems, where a direct approach would require quite a lot of
delicate accounting. Particular focus is given to the so-called
gambler's ruin problem, which is our first encounter with a
random walk.
Lecture 12
First we look at a concrete application of the gambler's ruin problem
in the field of medicine. Then we show that, given an event $F$ (with
$P(F)>0$), the conditional probability
$$
P(\cdot|F):2^S\to [0,1],\: E \mapsto P(E|F)
$$
is itself a probability on the sample space $S$ and can therefore be
further conditioned upon. Notice that $2^S$ denotes the collection of
all subsets of $S$.