WEBVTT
00:00:01.390 --> 00:00:02.400
Hi.
00:00:02.400 --> 00:00:05.140
In this problem, we're going to
use the set of probability
00:00:05.140 --> 00:00:07.830
axioms to derive the probability
of the difference
00:00:07.830 --> 00:00:09.170
of two events.
00:00:09.170 --> 00:00:11.430
Now, before we get started,
there's one thing you might
00:00:11.430 --> 00:00:13.730
notice that, the equation
we're trying to prove is
00:00:13.730 --> 00:00:15.340
actually quite complicated.
00:00:15.340 --> 00:00:17.730
And I don't like it either, so
the first thing we're going to
00:00:17.730 --> 00:00:21.930
do will be to find a simpler
notation for the events that
00:00:21.930 --> 00:00:23.180
we're interested in.
00:00:26.180 --> 00:00:29.780
So we start with two events, A
and B, and there might be some
00:00:29.780 --> 00:00:31.720
intersection between
the two events.
00:00:31.720 --> 00:00:36.460
We'll label the set of points
or samples in A that are not
00:00:36.460 --> 00:00:45.800
in B, as a set C. So C will be
A intersection B complement.
00:00:45.800 --> 00:00:51.680
Similarly, for all points that
are in B but not in A, this
00:00:51.680 --> 00:00:54.770
area, we'll call it D.
00:00:54.770 --> 00:01:01.310
And D will be the set A
complement intersection B. And
00:01:01.310 --> 00:01:06.130
finally, for points that are in
the intersection of A and
00:01:06.130 --> 00:01:13.540
B, we'll call it E. So E is A
intersection B. And for the
00:01:13.540 --> 00:01:15.830
rest of our problem, we're going
to be using the notation
00:01:15.830 --> 00:01:21.990
C, D, and E instead of
whatever's down below.
00:01:21.990 --> 00:01:26.010
If we use this notation, we can
rewrite our objective as
00:01:26.010 --> 00:01:27.790
the following.
00:01:27.790 --> 00:01:34.220
We want to show that the
probability of C union D is
00:01:34.220 --> 00:01:38.210
equal to the probability
of the event A plus the
00:01:38.210 --> 00:01:46.760
probability of B minus twice the
probability of E. And that
00:01:46.760 --> 00:01:48.010
will be our goal for
the problem.
00:01:51.140 --> 00:01:54.110
Now, let's take a minute to
review what the axioms are,
00:01:54.110 --> 00:01:56.310
what the probability
axioms are.
00:01:56.310 --> 00:01:59.000
The first one says
non-negativity.
00:01:59.000 --> 00:02:01.870
We take any event A, then
the probability of A
00:02:01.870 --> 00:02:04.150
must be at least 0.
00:02:04.150 --> 00:02:07.910
The second normalization says
the probability of the entire
00:02:07.910 --> 00:02:12.420
space, the entire sample space
omega, must be equal to 1.
00:02:12.420 --> 00:02:16.610
And finally, the additivity
axiom, which will be the axiom
00:02:16.610 --> 00:02:19.550
that we're going to use for this
problem says, if there
00:02:19.550 --> 00:02:23.780
are two events, A and B
that are disjoint--
00:02:23.780 --> 00:02:26.765
which means they don't have
anything in common, therefore.
00:02:26.765 --> 00:02:29.340
the intersection is
the empty set.
00:02:29.340 --> 00:02:32.870
Then the probability of their
union will be equal to the
00:02:32.870 --> 00:02:37.610
probably A plus the probability
of B. For the rest
00:02:37.610 --> 00:02:41.090
of the problem, I will refer
to this axiom as add.
00:02:41.090 --> 00:02:44.610
So whenever we invoke this
axiom, I'll write
00:02:44.610 --> 00:02:47.010
"add" on the board.
00:02:47.010 --> 00:02:48.510
Let's get started.
00:02:48.510 --> 00:02:52.950
First, we'll invoke the
additivity axioms to argue
00:02:52.950 --> 00:02:58.470
that the probability of C union
D is simply the sum of
00:02:58.470 --> 00:03:01.680
probability of C plus
probability of
00:03:01.680 --> 00:03:04.780
D. Why is this true?
00:03:04.780 --> 00:03:09.610
We can apply this axiom, because
the set C here and the
00:03:09.610 --> 00:03:14.410
set D here, they're completely
disjoint from each other.
00:03:14.410 --> 00:03:20.400
And in a similar way, we'll
also notice the following.
00:03:20.400 --> 00:03:30.390
We see that A is equal to the
union of the set C and E.
00:03:30.390 --> 00:03:36.020
And also, C and E, they're
disjoint with each other,
00:03:36.020 --> 00:03:39.970
because C and E by definition
don't share any points.
00:03:39.970 --> 00:03:44.440
And therefore, we have probably
A is equal to
00:03:44.440 --> 00:03:51.760
probability of C plus the
probability of E. Now, in a
00:03:51.760 --> 00:03:57.620
similar way, the probability of
event B can also be written
00:03:57.620 --> 00:04:03.890
as a probability of D plus the
probability of E, because
00:04:03.890 --> 00:04:07.480
event B is the union
of D and E.
00:04:07.480 --> 00:04:10.240
And D and E are disjoint
from each other.
00:04:10.240 --> 00:04:14.150
So we again invoke the
additivity axiom.
00:04:14.150 --> 00:04:18.529
Now, this should be enough
to prove our final claim.
00:04:18.529 --> 00:04:25.660
We have the probability of C
union D. By the very first
00:04:25.660 --> 00:04:29.810
line, we see this is simply
probability of C plus the
00:04:29.810 --> 00:04:33.690
probability of D.
00:04:33.690 --> 00:04:36.620
Now, I'm going to insert two
terms here to make the
00:04:36.620 --> 00:04:40.920
connection with a second part of
the equation more obvious.
00:04:40.920 --> 00:04:48.990
That is, I will write
probability C plus probability
00:04:48.990 --> 00:04:57.290
E plus probability D plus
probability of E. Now, I've
00:04:57.290 --> 00:04:59.810
just added two terms here--
00:04:59.810 --> 00:05:04.960
probability E. So to make the
equality valid or subtract it
00:05:04.960 --> 00:05:10.100
out two times, the
probability of E.
00:05:10.100 --> 00:05:13.510
Hence this equality is valid.
00:05:13.510 --> 00:05:17.370
So if we look at this equation,
we see that there
00:05:17.370 --> 00:05:20.260
are two parts here that
we've already seen
00:05:20.260 --> 00:05:22.810
before right here.
00:05:22.810 --> 00:05:30.570
The very first parenthesis is
equal to the probability of A.
00:05:30.570 --> 00:05:34.690
And the value of the second
parenthesis is equal to the
00:05:34.690 --> 00:05:39.750
probability of B. We just
derived these here.
00:05:39.750 --> 00:05:47.730
And finally, we have the minus
2 probability of E. This line
00:05:47.730 --> 00:05:51.450
plus this line gives us
the final equation.
00:05:51.450 --> 00:05:53.610
And that will be the answer
for the problem.