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## Compound probability of independent events using the multiplication rule

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# Probability without equally likely events

## Video transcript

So far, we've been dealing
with one way of thinking about probability, and
that was the probability of A occurring is the
number of events that satisfy A over all of the
equally likely events. And this is all of the
equally likely events. And so in the case
of a fair coin, the probability of heads--
well, it's a fair coin. So there's two
equally likely events, and we're saying one of
them satisfies being heads. So there's a 1/2 chance
of you having a heads. The same thing for tails. If you took a die, and
you said the probability of getting an even number
when you roll the die. Well, there's six
equally likely events, and there's three even
numbers you could get. You could get 2, a 4, or a 6. So there's three even numbers. So once again, you have a
1/2 chance of that happening. And this is a really
good model where you have equally likely
events happening. Now I'm going to change
things up a little bit. So I'm going to draw a line
here because this was just one way of thinking
about probability. Now we're going to
introduce another one that's more helpful when we can't think
about equally likely events. And in particular, I'm going
to set up an unfair coin. So this right over here is
going to be my unfair coin. So that is my coin. Well, I could draw the coin. So it's a gold coin this time. It is unfair. One side of that coin is a
little heavier than the other, even though it's
meant to look fair. So it still has that picture
of some president or something on one side of it. So this is the head side. This is heads, and then,
obviously, on the back, you have tails. But as I mentioned,
this is an unfair coin. And I'm going to make
it interesting statement about this unfair coin and
one that really doesn't fit into the mold that
I set up over here, and this interesting
statement is that we have more than a 50/50
chance of getting heads or more than a 50% chance
or more than a 1/2 chance of getting heads. I'm going to say that the
probability of getting heads for this coin right
over here is 60%. Or another way to
say it, it's 0.6. Or another way to say
it, it is 6 out of 10. Or another way to
say it, it is 3/5. And this might make
intuitive sense to you and hopefully it
does a little bit, but I want you to
realize that this is fundamentally different
than what we were saying before because now we can't say that
there are two equally likely events. There are two possible events. You can either get
heads or tails. We're assuming that the
coin won't fall on its edge. That's impossible. So you're either going
to get heads or tails, but they're not
equally likely anymore. So we really can't do
this kind of counting the number of events
that satisfy something over all of the possible events. In this situation, in order
to visualize the probability, we have to kind of take what's
called a "frequentist approach" or think about it in terms
of frequency probability. And the way to conceptualize
a 60% of getting heads is to think, if we had a
super large number of trials, if we were to just flip
this coin a gazillion times, we would expect that 60% of
those would come up heads. It's unclear how I
determined that this is 60%. Maybe I ran a
computer simulation. Maybe I know exactly all
of the physics of this, and I could completely model how
it's going to fall every time. Or maybe I've actually
just run a ton of trials. I've flipped the coin a
million times, and I said, wow, 60% of those, 600,000
of those, came up heads. And then, we could make a
similar statement about tails. So if the probability
of heads is 60%, the probability
of tails-- well, there's only two
possibilities, heads or tails. So if I say the probability
of heads or tails, it's going to be equal
to 1 because you're going to get one of
those two things. You have 100% chance of
getting a heads or a tails, and these are mutually
exclusive events. You can't have both of them. The probability
of tails is going to be 100% minus the
probability of getting heads, and this, of course, is 60%. So it's 100% minus 60%, or
40%, or as a decimal, 0.4, or as a fraction, 4/10, or as
a simplified fraction, 2/5. So, once again,
this probability is saying-- we can't say
equally likely events. We could say that,
if we're going to do a gazillion of
these, we would expect, as we get more and more and more
trials, more and more flips, 40% of those would be heads. Now, with that out of
the way, let's actually do some problems with this. So let's think about the
probability of getting heads on our first flip and
heads on our second flip. So, once again, these
are independent events. The coin has no memory. Regardless of what I
got on the first flip, I have an equal chance
of getting heads on the second flip. It doesn't matter if I got
heads or tails on the first. So this is the probability of
heads on the first flip times the probability of heads
on the second flip, and we already know. The probability of heads on
any flip is going to be 60%. I'll write it as a decimal. It makes the math a little
bit easier, 0.6, 0.6, and we can just multiply. I'll do it right over here. So this is 0.6 times 0.6. Now, it's always good
to do a reality check. One way to think about it
is I'm taking 6/10 of 6/10, so it should be a little
bit more than half of 6/10 or probably a little
bit more than 3/10. And we've explain
this in detail where we talk about
multiplying decimals, but we essentially just
multiply the numbers, not thinking about
the decimals at first. 6 times 6 is 36. And then you count
the number of digits we have to the right
of the decimal. We have one, two to the
right of the decimal. So we're going to
have two to the right of the decimal in our answer. So it is 0.36, and
that makes sense. We're taking 60% of 0.6. We're taking 0.6 of 0.6, a
little bit more than half of 0.6. And, once again, it's a
little bit more than 0.3. So this also makes sense. So it's 0.36. Or another way to
think about it is there's a 36%
probability that we get two heads in a row,
given this unfair coin. Remember, if it was a fair
coin, it would be 1/2 times 1/2, which is 1/4, which
is 25%, and it makes sense that this is more than that. Now, let's think about a
slightly more complicated example. Let's say the probability
of getting a tails on the first flip, getting
a heads on the second flip, and then getting a
tails-- I'm going to do this in a new
color-- and then getting a tails on the third flip. So this is going to be equal
to the probability of getting a tails on the first flip
because these are all independent events. If you know that you had
a tail on the first flip, that doesn't affect
the probability of getting a heads
on the second flip. So times the probability
of getting a heads on the second flip,
and then that's times the probability of getting
a tails on the third flip. And the probability of getting
a tails on any flip we know is 0.4. The probability of getting
a heads on any flip is 0.6, and then the probability of
getting tails on any flip is 0.4. And so, once again, we
can just multiply these. So 0.4 times 0.6. There's actually a couple of
ways we can think about it. Well, we could literally say,
look, we're multiplying 4 times 6 times 4, and then
we have three numbers behind the decimal point. So let's do it that way. 4 times 6 is 24. 24 times 4 is 96. So we write a 96,
but remember, we have three numbers
behind the decimal point. So it's one to the right
of the decimal there, one to the right of
the decimal there, one to the right
of decimal there. So three to the right. So we need three to the right
of the decimal in our answer. So one, two-- we need one more
to the right of the decimal. So our answer is 0.096. Or another way to think about it
is-- write an equal sign here-- this is equal to a 9.6% chance. So there's a little bit
less than 10% chance, or a little bit less
than 1 in 10 chance, of, when we flip this
coin three times, us getting exactly a
tails on the first flip, a heads on the second flip,
and a tails on the third flip.