Let us summarize our findings today.
Summary of Tableau 8, Part 1.
Characterising side information in chance experiments, and
the role of conditional probability.
Side information in our chance experiments, is always folded into
the fabric by the specification of the occurrence of an ancillary event.
It's A, B.
We define the conditional probability of
A given that an event B of positive probability has occurred.
To be inwards the proportional part of A which resides in B,
and formally as a math, as a matter of mathematical definition.
The ratio of the probability of the intersection of A and
B to the probability of the ancillary of A to B.
It's a very clean and simple definition,
this is something that you should remember.
From the definition arises another piece of insight.
The definition of conditional probability is endelectibly intwined with
the idea of conjuctional probabilities.
So we could write down conjunction probabilities,
probabilities of intersections, in terms of a chain of conditional probabilities.
Either things of this kind are occasionally useful,
when it transpires that the problem setting makes computations of conditional
probabilities straightforward, where the computation of conjunction probabilities
might not be quite so simple, might be more opaque.
This is now the of conditional probability that we will deal with.
In our next tableau, we will return to our guiding principle of additivity,
one of the pillars on which this entire science of probability rests.
And we shall see how
conditional probability puts this principle in a new and beguiling light.
We shall discover a panoply of applications.
Subtle, deep but elementary.
Arising out of this very simple exercise of this principle.
So this will be on tap for our next lecture.