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# Dependent probability example

## Video transcript

Let's do another one of these
dependent probability problems. You have 4 coins in a bag. 3 of them are
unfair in that they have a 45% chance of coming
up tails when flipped. The rest are fair. So for the rest of them, you
have a 50% chance of tails or a 50% chance of heads. You randomly choose
one coin from the bag and flip it 4 times. What is the percent
probability of getting 4 heads? So let's think about it. When we put our hand in the bag
and we take one of the coins out, there's some probability
that we get an unfair coin. And 3 of the 4 coins are unfair. So there's a 3/4 probability
that we get an unfair coin. And then there is only 1 out
of the 4 coins that's fair. So there was a 1/4 probability
that I get a fair coin. So unfair, let's
remind ourselves-- an unfair coin has a 45%
chance of coming up tails. So this means that I have a
45% chance of tails, which also means-- and we have
to be careful here because they're asking us
about heads-- if I have a 45% chance of
getting tails, that means I have a 55%
chance of getting heads. I have a 100% chance of
getting one of these two. If it's 45% for tails, 100
minus 45 is 55 for heads. For the fair coin, I have
a 50% chance of tails and a 50% chance of heads. 50% heads. Fair enough. Now I want to know, in
either of these scenarios, what is the percent probability
of getting four heads? So given I've got
the unfair coin, the probability of
getting four heads is going to be 55% for
each of those flips. So the probability of
getting exactly four heads is going to be 0.55 times
0.55 times 0.55 times 0.55. And so the probability
of picking an unfair coin and getting four
heads in a row is going to be equal to 3/4 times
all of this business over here. So that's 3/4 times--
and this is 0.55 times itself four times, so I
could write it as 0.55 to the fourth power. And we'll get the
calculator out in a second to actually calculate
what this is. Now let's do the same
thing for the fair coin. If I did pick a fair
coin, the probability of getting heads four times in
a row is going to be 0.5 times 0.5 times 0.5 times 0.5. Or the probability of
getting the fair coin, which is 1/4 chance, times
the probability-- and getting four heads in a
row is going to be 1/4 times all of this. So it's going to be 1/4 times--
this is just 0.5 times itself four times, so that's
0.5 to the fourth power. So let's get the
calculator out to calculate either one of these. So we get 3 divided
by 4 times-- and it knows that when I do
the multiplication, it's not in the
denominator here. So it's 3/4 times-- and I'll
just do it in parentheses, which I don't have to
do it in parentheses, because it knows
order of operations-- so 0.55 to the fourth
power is equal to 0.-- let me write it down. Let me take it off the screen
so I can write it down properly. Actually, let me just do
both of these calculations. So this probability is
that one right over there. And then this one down here
is 1 divided by 4 times 0.5 to the fourth power. So it's equal to that
right over there. And so let's be clear. The probability of
picking the unfair coin and then getting four heads
in a row is this top number. It's like roughly 6.9 chance
that you get the unfair coin and then get four
heads in a row. The probability that
you get the fair coin and then get four heads
in the row is even lower. It's only a 1.6% chance. Now the probability of getting
four heads in a row either way is going to be the
sum of this and this, or the sum of that
and that, which is going to be-- let me
keep my calculator out-- so it's going to be
equal to-- I can just take the previous answer. Let me just retype it
so I don't confuse you. So 0.015625 plus 0.0686296875. I'm going to round it anyway,
so it won't matter too much. So if I take the sum-- let
me take this off screen so I can still see it
and then let me write it. So what I got here,
this one is 0.068629. And I'll round it, 7. This down here was 0.015625
and when you add these two up-- because we just care about
getting four heads either way. There's a probability
of getting it this way with the unfair coin. This is the probability of
getting it with a fair coin. We want it either way. So let's add the two, which we
already did on our calculator. So if you add that
number to that number. You get 0.08425
and it keeps going, but I'm just going to round it. This is equal to
8.425%, or if I want to round it a little
bit more, 8.43% chance of getting
four heads in a row. And once again, that's
a slightly higher number than if all of the
coins were fair. Because there's a
3/4 chance that I get a coin that has a better
than even chance of getting heads. So that's why this number is
going to be a little bit higher than the probability if I had
a fair coin, of just getting four heads in a row.