In this lecture, we introduce and develop the concept of
independence between events.
The general idea is the following.
If I tell you that a certain event A has occurred, this
will generally change the probability of some other
event B. Probabilities will have to be replaced by
conditional probabilities.
But if the conditional probability turns out to be
the same as the unconditional probability, then the
occurrence of event A does not carry any useful information
on whether event B will occur.
In such a case, we say that events A and B are
independent.
We will develop some intuition about the meaning of
independence of two events and introduce an extension, the
concept of conditional independence.
We will then proceed to define the independence of a
collection of more than two events.
If, for any two of the events in the collection we have
independence between them, we will say that we have pairwise
independence.
But we will see that independence of the entire
collection is something different.
It involves additional conditions.
Finally, we will close with an application in reliability
analysis and with a nice puzzle that will serve as a
word of caution about putting together probabilistic models.