Basic Conditional Probability
Overview
Conditional probability deals with situations where a probability space has been defined and there is additional information
which constrains the space.
As an example, consider the situation of rolling two fair six sided die. One can formulate a set of outcomes and assign probabilities.
Now, assume that one is told that the sum of the two die is equal to four. This represents new information, and any probabilistic statements
made using the formulated probability space would be inaccurate wihtout accounting for the the given information.
Information Event
When additional information is obtained about a given probability space, this information is typically represented as
some event having occurred.
For example, when the sum of two die is known to be four, this means that only the following sample points are
possible outcomes
Event = {{1,3}, {2,2}, {3,1}}
This set of sample points is a subset of all possible points and there is an event.
Conditional Probabilities
{% P(A|B) = \frac{P(A \cap B)}{P(B)} %}
Conditional Probabilities
The conditional probability framework defines a new set of probabilities by redefining the probability space so that the
information event (given above) becomes the new universe of possible outcomes. In order to ensure that the probability of
entire space is now equal to 1, a new set of probabilities are assigned to every point.
Probabilities are scaled up proportionately, in the discrete case we have
{% P(X = x_i | Y = y_j) = \frac{P(X=x_i, Y = y_j)}{P(Y=y_j)} %}
In the continuous case, the
density function
becomes
{% f(x|y) = \frac{f_{xy}(x,y)}{f_y(y)} %}
