We will now go through a probability calculation for
the case where we have a continuous sample space.
We revisit our earlier example in which we were throwing a
dart into a square target, the square target
being the unit square.
And we were guaranteed that our dart would fall somewhere
inside this set.
So our sample space is the unit square itself.
We have a description of the sample space, but we do not
yet have a probability law.
We need to specify one.
The choice of a probability law could be arbitrary.
It's up to us to choose how to model a certain situation.
And to keep things simple, we're going to assume that our
probability law is a uniform one, which means that the
probability of any particular subset of the sample space is
going to be the area of that subset.
So if we have some subset lying somewhere here and we
ask what is the probability that we fall into that subset?
The probability is exactly the area of
that particular subset.
Once more, this is an arbitrary choice of a
probability law.
There's nothing in our assumptions so far that would
force us to make this particular choice.
And we just use it for the purposes of this example.
So now let us calculate some probabilities.
Let us look at this event.
This is the event that the sum of the two numbers that we get
in our experiment is less than or equal to 1/2.
It is always useful to work in terms of a picture and to
depict that event in a picture of the sample space.
So in terms of that sample space, the points that make
this event to be true are just a triangle that lies below the
line, where this is the line, that's x plus y equals 1/2.
Anything below that line, these are the outcomes that
make this event happen.
So we're trying to find the probability of this red event.
We have assumed that probability is equal to area.
Therefore, the probability we're trying to calculate is
the area of a triangle.
And the area of a triangle is 1/2 times the base of the
triangle, which is 1/2 in our case, times the height of the
triangle, which is again 1/2 in our case.
And the end result is 1/8.
Let us now calculate another probability.
Now, this is an event that consists of
only a single element.
We take the point 0.5, 0.3, which sits somewhere here.
The event of interest is a set, but that set consists of
a single point.
So we're asking for the probability that our dart
falls exactly on top of that point.
What is it?
Well, it is the area of a set that
consists of a single point.
What is the area of a single point?
It is 0.
And similarly for any other single point inside that
sample space that we might have considered, the answer is
going to be 0.
Let us now abstract from this example, as well as the
previous one, and note the following.
Probability calculations involve a
sequence of four steps.
Starting with a word description of a problem, of a
probabilistic experiment, we first write
down the sample space.
Then we specify a probability law.
Let me emphasize again here that this step has some
arbitrariness in it.
You can choose any probability law you like, although for
your results to be useful it would be good if your
probability law captures the real-world phenomenon you're
trying to model.
Typically you're interested in calculating the probability of
some event.
That event may be described in some loose manner, so you need
to describe it mathematically.
And if possible, it's always good to describe it in terms
of a picture.
Pictures are immensely useful when going
through this process.
And finally, the last step is to go ahead and calculate the
probability of the event of interest.
Now, a probability law in principle specifies the
probability of every event, and there's
nothing else to do.
But quite often the probability law will be given
in some implicit manner, for example, by specifying the
probabilities of only some of the events.
In that case, you may have to do some additional work to
find the probability of the particular event
that you care about.
This last step sometimes will be easy.
Sometimes it may be complicated.
But in either case, by following this four-step
procedure and by being systematic you will always be
able to come up with a single correct answer.