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You have eight coins in a bag. Three of them are
unfair in that they have a 60% chance of coming
up heads when flipped. The rest are fair coins. So if three are unfair,
the rest are fair coins. When this problem says
that they are fair coins, it means that they have a
50/50 chance of coming up either heads or tails. You randomly choose
one coin from the bag and flip it two times. What is the percent probability
of getting two heads? So this is an
interesting question, but if we break it
down, essentially with a decision tree,
it'll help break it down a little bit better. So let's say that we have a
bag, three of them are unfair. So we could even
visualize a bag. You don't have to do
this all the time. I'll do the fair coins in white. One, two, three, four,
five fair coins, and we have three unfair coins. One, two, three. And this whole thing is
my bag, right over here. That is my bag of coins. When I take my
hand in, if I were to take any of
these white coins, there's a 50% chance that
it gets heads on any flip. The odds of getting two heads
in a row would be 50% times 50% for these white coins. But I don't know I'm
going to get a white coin. If I get one of
these orange coins, I have a 60% chance
of coming up heads. If I have picked one
of these orange coins, the probability of
getting heads twice is going to be 60% times 60%. So how do I factor
in this idea that I don't know if I've
picked a white fair coin or an orange unfair coin. We'll assume that
the coins actually aren't white and orange. They all look like
regular coins. So what I'll do is I'll draw a
little bit of a decision tree here. I guess maybe I could call
it a probability tree. So there's some probability
that I pick a fair coin. And there's some probability
that I pick an unfair coin. And so what is the probability
that I pick a fair coin? Well, one, two, three, four,
five out of the total eight coins are fair, so there
is a 5/8 probability. I'll write it here on
the branch, actually. So there's a 5/8 chance
that I pick a fair coin, and then there is a 3-- one,
two, three, out of 8 chance that I pick an unfair coin. So if I were to just
tell you, what's the probability of
picking a fair coin? You'd say oh, 5/8. What's the probability
of an unfair coin? 3/8. And you could convert that
to a decimal or a percentage or whatever you'd like. Now, given that I have
picked a fair coin, what is the probability
that I will get heads twice? So let me write it this way. Now this is just
notation right here. So the probability of--
I'll call it heads heads-- so I get two heads
in a row, given that I have a fair coin-- it
looks like very fancy notation, but it's just like look,
if you knew for a fact that coin you had is
absolutely fair-- that it has a 50% chance of
coming up heads-- what is the probability of
getting two heads in a row? Then we can just say,
well, that's just going to be 50%, so 50% times
50%, which is equal to 25%. Which is equal to 25%. What is the probability
that you picked a fair coin and you got two heads in a row? So given that you
have a fair coin, it's a 25% chance that you
have two heads in a row. But the probability of
picking a fair coin and then given the fair coin
getting two heads in a row, will be the 5/8 times the 25%. So this whole branch--
I should maybe draw it this way--
the probability of this whole series
of events happening. So starting with you
picking the fair coin and then getting
two heads in a row will be-- I'll
write it this way-- it will be 5/8 times this right
over here, times the 0.25. I want to make it very clear. The 0.25 is the
probability of getting two heads in a
row given that you knew that you got a fair coin. But the probability
of this whole series of events happening,
you would have to multiply this times the
probability that you actually got a fair coin. So another way of
thinking about it is this is the probability
that you got a fair coin and that you have
two heads in a row. Now let's do the same
thing for the unfair coin. So the probability-- I'll do
that in the same green color-- the probability that
I get heads heads given that my coin is unfair. So if you were to somehow
know that your coin is unfair, what is the probability of
getting two heads in a row? Well in this unfair
coin, it has a 60% chance of coming up heads. So it will be equal to 0.6
times 0.6, which is 0.36. If you have an unfair coin--
if you know for a fact that you have an unfair
coin, if that is a given-- you have a 36% chance of
getting two heads in a row. Now if you want to
know the probability of this whole series of
events-- the probability that you picked an
unfair coin and you get two heads in a row, so
the probability of unfair and two heads in a row given
that you had that unfair coin-- you would multiply this
3/8 times the 0.36. So this will be equal
to 3/8 times 0.36. And so let's get a calculator
out and calculate these. So if I take 5 divided by 8
times 0.25, I get 0.15625. So this is equal to 0.15625. And then if I do the
other part, so if I have 3 divided by 8 times 0.36,
that gives me 0.135. So this is 0.135. So if someone were
to ask you, what's the probability of
picking the fair coin and then getting two heads
in a row with that fair coin, you would get this number. If someone were to say,
what's the probability of you picked the unfair
coin and then get two heads in a row
with that unfair coin, you would get this number. Now, if someone were
to say, either way, what's the probability of
getting two heads in a row? Because that's what
they're asking us here. What is the probability
of getting two heads? So we could get it through this
method-- by chance, picking the fair coin-- or through this
method-- by chance, picking the unfair coin. So since we can
do it either way, we can sum up the probabilities. Either of these events
meet our constraints. So we can just add
these two things up. So let's do that. So we can add 0.135 plus
0.15625 gives us 0.29125. So point 0.29125, that's when
we add 0.15625 plus 0.135 will equal this. And if we want to write
it as a percentage, you essentially just
multiply this times 100 and add the
percentage sign there. So this is equal to 29.125%. Or if we were to round to
the nearest hundredths, then this would be
the exact number, or we could say it's
approximately 29.13%, depending on how much
we need to round it. So we have a little less than
a 1/3 chance of this happening. And the reason why-- if
everything in the bag was a fair coin, there'd be a
25% chance of this happening because you just say, OK any
of these, they're all the same. Flip it twice, 25% chance. Our chance is a
little bit higher because there's some
probability-- there's a 3/8 chance-- that we pick a coin
that has a higher than even chance of coming up heads.