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Addition rule for probability | Probability and Statistics | Khan Academy - YouTube
Channel: Khan Academy
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Let's say I have a bag.
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And in that bag--
I'm going to put
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some green cubes in that bag.
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And in particular, I'm going
to put eight green cubes.
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I'm also going to put
some spheres in that bag.
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Let's say I'm going
to put nine spheres.
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And these are the green spheres.
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I'm also going to put some
yellow cubes in that bag.
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I'm going to put five of those.
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And I'm also going to put some
yellow spheres in this bag.
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And let's say I
put seven of those.
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I'm going to stick
them all in this bag.
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And then I'm going
to shake that bag.
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And I'm going to pour it out.
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And I'm going to look
at the first object that
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falls out of that bag.
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And what I want to think
about in this video
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is what are the
probabilities of getting
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different types of objects?
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So for example, what
is the probability
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of getting a cube of any color?
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What is the probability
of getting a cube?
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Well, to think about that
we should think about what--
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or this is one way
to think about it--
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what are all of the equally
likely possibilities that
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might pop out of the bag?
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Well, we have 8 plus 9 is 17.
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17 plus 5 is 22.
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22 plus 7 is 29.
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So we have 29 objects.
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There are 29 objects in the bag.
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Did I do that right?
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This is 14, yup 29 objects.
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So let's draw all of
the possible objects.
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I'll represent it as this
big area right over here.
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So these are all the
possible objects.
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There are 29 possible objects.
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So there's 29
equal possibilities
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for the outcome of my experiment
of seeing what pops out
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of the bag, assuming
that it's equally
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likely for a cube or a
sphere to pop out first.
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And how many of them meet our
constraint of being a cube?
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Well, I have eight green cubes,
and I have five yellow cubes.
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So there are a
total of 13 cubes.
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So let me draw
that set of cubes.
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So there's 13 cubes.
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We could draw it like
this-- there are 13 cubes.
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This right here is the
set of cubes, this area.
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And I'm not drawing it exact.
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I'm approximating.
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It represents the
set of all the cubes.
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So the probability
of getting a cube
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is the number of events
that meet our criteria.
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So there's 13
possible cubes that
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have an equally likely
chance of popping out,
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over all of the possible equally
likely events, which are 29.
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That includes the
cubes and the spheres.
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Now let's ask a
different question.
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What is the probability of
getting a yellow object, either
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a cube or a sphere?
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So once again, how many things
meet our conditions here?
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Well, we have 5 plus 7.
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There's 12 yellow
objects in the bag.
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So we have 29 equally
likely possibilities.
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I'll do it in that same color.
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We have 29 equally
likely possibilities.
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And of those, 12
meet our criteria.
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So let me draw 12
right over here.
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I'll do my best attempt.
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So let's say it looks
something like-- so
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the set of yellow objects.
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There are 12 objects
that are yellow.
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So the 12 that
meet our conditions
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are 12, over all the
possibilities-- 29.
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So the probability
of getting a cube--
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13 29ths, probability of
getting a yellow-- 12 29ths.
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Now let's ask something a
little bit more interesting.
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What is the probability
of getting a yellow cube?
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So I'll put it in yellow.
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So we care about the color, now.
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So this thing is yellow.
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What is the probability of-- or
as my son would say, "lello."
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What is the probability
of getting a yellow cube?
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Well, there's 29 equally
likely possibilities.
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And of those 29 equally likely
possibilities, 5 of those
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are yellow cubes, or
"lello" cubes, five of them.
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So the probability is 5 29ths.
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And where would we see
that on this Venn diagram
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that I've drawn?
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This Venn diagram is
just a way to visualize
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the different probabilities.
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And they become
interesting when you
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start thinking about
where sets overlap,
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or even where they
don't overlap.
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So here we are
thinking about things
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that are members
of the set yellow.
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So they're in this set,
and they are cubes.
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So this area right
over here-- that's
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the overlap of these two sets.
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So this area right
over here-- this
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represents things that
are both yellow and cubes,
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because they are
inside both circles.
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So this right over here-- let
me rewrite it right over here.
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So there's five objects that
are both yellow and cubes.
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Now let's ask-- and this is
probably the most interesting
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thing to ask-- what is
the probability of getting
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something that is yellow or or
a cube, a cube of any color?
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The probability of getting
something that is yellow
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or a cube of any
color-- well, we still
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know that the denominator
here is going to be 29.
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These are all of the
equally likely possibilities
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that might jump out of the bag.
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But what are the possibilities
that meet our conditions?
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Well, one way to think about
it is, well, the probability--
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there's 12 things that would
meet the yellow condition.
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So that would be this entire
circle right over here-- 12
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things that meet the
yellow condition.
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So this right over here is 12.
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This is the number of yellow.
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That is 12.
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And then to that, we can't
just add the number of cubes,
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because if we add
the number of cubes,
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we've already counted these 5.
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These 5 are counted
as part of this 12.
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One way to think
about it is there
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are 7 yellow objects
that are not cubes.
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Those are the spheres.
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There are 5 yellow
objects that are cubes.
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And then there are 8
cubes that are not yellow.
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That's one way to think about.
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So when we counted this
12-- the number of yellow--
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we counted all of this.
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So we can't just add
the number of cubes
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to it, because then we would
count this middle part again.
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So then we have to
essentially count
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cubes, the number of
cubes, which is 13.
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So the number of
cubes, and we'll
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have to subtract out this
middle section right over here.
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Let me do this.
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So subtract out the middle
section right over here.
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So minus 5.
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So this is the number
of yellow cubes.
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It feels weird to write
the word yellow in green.
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The number of yellow cubes-- or
another way to think about it--
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and you could just do
this math right here.
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12 plus 13 minus 5 is 20.
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Did I do that right?
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12 minus, yup, it's 20.
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So that's one way.
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You just get this is
equal to 20 over 29.
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But the more interesting
thing than even the answer
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of the probability
of getting that,
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is expressing this in terms
of the other probabilities
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that we figured out
earlier in the video.
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So let's think about
this a little bit.
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We can rewrite this
fraction right over here.
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We can rewrite this as 12 over
29 plus 13 over 29 minus 5
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over 29.
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And this was the
number of yellow
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over the total possibilities.
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So this right over here
was the probability
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of getting a yellow.
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This right over
here was the number
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of cubes over the
total possibilities.
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So this is plus the
probability of getting a cube.
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And this right over
here is the number
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of yellow cubes over
the total possibilities.
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So this right over here
was minus the probability
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of yellow, and a cube.
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I'm not going to
write it that way.
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Minus the probability
of yellow-- I'll
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write yellow in yellow--
yellow and a cube.
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And so what we've
just done here--
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and you could play
with the numbers.
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The numbers I just used
as an example right
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here to make things a
little bit concrete.
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But you can see this is
a generalizable thing.
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If we have the probability
of one condition
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or another condition-- so let me
rewrite it-- the probability--
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and I'll just write it a
little bit more generally here.
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This gives us an
interesting idea.
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The probability of getting
one condition of an object
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being a member of set
a, or a member of set b
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is equal to the probability
that it is a member of set a,
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plus the probability that
is a member of set b,
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minus the probability
that is a member of both.
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And this is a really
useful result.
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I think sometimes it's
called the addition
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rule of probability.
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But I want to show you that
it's a completely common-sense
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thing.
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The reason why you can't just
add these two probabilities
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is because they might
have some overlap.
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There's a probability
of getting both.
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And if you just
added both of these,
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you would be double
counting that overlap,
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which we've already seen
earlier in this video.
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So you have to subtract
one version of the overlap
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out so you are not
double counting it.
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I'll throw another
one other idea out.
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Sometimes, you
have possibilities
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that have no overlap.
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So let's say this is the
set of all possibilities.
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And let's say this is the
set that meets condition a
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and let me do this
in a different color.
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And let's say that this is the
set that meets condition b.
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So in this situation,
there is no overlap.
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There's no way-- nothing is a
member of both sets, a and b.
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So in this situation, the
probability of a and b is 0.
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There is no overlap.
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And these type of conditions,
or these two events,
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are called mutually exclusive.
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So if events are
mutually exclusive,
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that means that they both
cannot happen at the same time.
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There's no event that meets
both of these conditions.
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And if things are
mutually exclusive,
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then you can say the
probability of a or b
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is the probability of a plus
b, because this thing is 0.
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But if things are not
mutually exclusive,
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you would have to
subtract out the overlap.
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And probably the
best way to think
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about it is to
just always realize
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that you have to
subtract out the overlap.
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And obviously if something
is mutually exclusive,
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the probability of getting
a and b is going to be 0.
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