One last example pays for all.
Additivity once more.
So let's begin with a setting, an abstract setting.
Of course, we have a sample space, omega.
Define a partition of omega to be any finite collection of events.
Say, A1, A2, A3 through a finite, all countably infinite collection.
Right. And
we will denote it in notation by simply a set of events, A with a subscript j or
a j, for example, runs through the natural numbers.
Either a finite range or all the natural numbers.
This is called a partition of omega.
If the events Aj are mutually disjoint,
the intersection of any two of them is empty.
And the events are so constituted that together,
they make up the entire sample space.
Right? In other words, the union of all these Ajs
is omega and this is sort of visually indicated in the cuts,
the partitions I've showed you on the right in your figure.
The moment we have this,
additivity immediately gives rise to the following identity.
Right. This is the axiom of additivity for
probability measure and we're going to apply it to the union of all these ajs.
By normalization, the sample space has got unit probability, but
the sample space may be decomposed into a disjoint union of a finite or
countably infinite collection of events.
And so by additivity,
this must be equal to the sum of the probabilities of these events.
And so here is a conclusion.
If one has a partition of the sample space, then the sum of
the probabilities of the members of the partition must be one.
This is a powerful, general idea in principle.
Again, this is additivity to the four.
But this principle at its heart, its great utility, rests on the idea,
this philosophical idea that we can take a complex problem and break it up into
pieces, which have got individual characters, which you can then analyze.
And then again, just stitch everything together by adding up all the pieces.
Very powerfu, general principle.
A particular manifestation of this is something we've already seen and
turns out to be quite remarkably useful.
Consider a very special case, where we partition omega into two pieces.
A set A and what remains?
Well, of course, the complement of A.
Clearly, the sets A and A complement are a partition of omega.
A and A complement, share no elements.
They are disjoint and the union is all of omega.
Our additivity principle immediately tells us then, that the some of
the probabilities of A and the complement of A must indeed perforce be unit.
We use this repeatedly in our elementary examples.
And I can't emphasize enough how useful and important this observation is.