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## Counting principle and factorial

## Video transcript

Let's say that I've just
won some type of contest at a car dealership,
and they're going to give me a brand new car. And in deciding which
car they give me, they're first going to randomly
select the engine type. So the engine will come in
two different varieties. It'll either be a four-cylinder
or a six-cylinder engine. And they're literally just
going to flip a fair coin to decide whether I get
a four-cylinder engine or a six-cylinder engine. Then they're going
to pick the color. And there's four different
colors that the cars come in. So I'll write color
in a neutral color. So you could get a red car. That's not red. Let me do that in actual
red color, or close to red. You could get a red car,
you could get a blue car, you could get a green car,
or you could get a white car. And once again, they're going
to have the red, blue, green, and white in little
slips of paper in a bowl and they're just going
to pick one of them out. So all of these
are equally likely. So given this, that
they're just going to flip a coin to
pick the engine, and that all of these, the
colors all equally likely, I want to think
about the probability of getting a
six-cylinder white car. So I encourage you
to pause the video and think about it on your own. Well, one way to
think about this is, well, what are
all the equally likely possible outcomes? And then which of those
match six-cylinder white car? Well, first, we could think
about the engine decision. We're either going to get
a four-cylinder engine. So the first decision
is the engine. You could view it that way. You're either going to get
a four-cylinder engine, or you're going to get
a six-cylinder engine. Now, if you got a
four-cylinder engine, you're either going to get
red, blue, green, or white. And if you've got a
six-cylinder engine, once again, you're
either going to get red, blue-- I think you see
where this is going. That's not blue. Red, blue, green, or white. So how many possible
outcomes are there? Well, you could just count. You could kind of
say, the leaves of this tree diagram-- one, two,
three, four, five, six, seven, eight possible outcomes. And that makes sense. You have two possible engines
times four possible colors. You see that right
here-- one group of four, two groups of four. So this outcome right here
is a four-cylinder blue car. And this outcome over here
is a six-cylinder green car. So there's eight equally
possible outcomes. And which outcome matches
the one that we, I guess, are hoping for, the
white six-cylinder car? Well, that's this
one right over here. It's one of eight
equally likely events. So we have a 1/8 probability. This wasn't the only
way that we could have drawn the tree diagram. We could have
thought about color as the first row of this tree. So we could have said,
look, we're either going to get a-- let
me do it down here, so I have a little more
space-- we're either going to get a red, a
blue-- that's not blue. Changing colors
is the hard part. A blue, a green, or a white car. And then for each
of those colors, I'm either going to get a
four-cylinder or a six-cylinder engine. So it's either going
to be four or six. This would be another way
of drawing a tree diagram to represent all
of the outcomes. So what is this outcome
right over here? This is a six-cylinder red car. This is a four-cylinder
blue car right over here. Which is the one
that we care about? White six-cylinder car? That's this outcome
right over here. Once again, you see you have
eight equally likely outcomes. And that happens because you
have four possible colors. And for each of those
four possible colors, you have two different
engine types.