Main content

## Multiplication rule for dependent events

Current time:0:00Total duration:6:38

# Dependent probability introduction

AP Stats: VAR‑4 (EU), VAR‑4.D (LO), VAR‑4.D.2 (EK)

## Video transcript

Let's imagine
ourselves in some type of a strange casino
with very strange games. And you walk up to a
table, and on that table there is an empty bag. And the guy who
runs the table says, look, I've got
some marbles here, 3 green marbles,
2 orange marbles. And I'm going to
stick them in the bag. And he literally sticks
them into the empty bag to show you that it's
truly 3 green marbles and 2 orange marbles. And he says, the game
that I want you to play, or if you choose
to play, is you're going to look away-- stick
your hand in this bag, the bag is not transparent--
feel around the marbles. All the marbles feel
exactly the same. And if you're able to
pick 2 green marbles, if you're able to take 1 marble
out of the bag, it's green. You put it down on the table. Then put your hand back in the
bag, and take another marble. And if that one's also
green, then you're going to win the prize. You're going to win $1
if you get 2 greens. If you get 2 greens,
you're going to win $1. You say, well, this sounds
like an interesting game. How much does it cost to play? And the guy tells you,
it is $0.35 to play, so obviously a fairly
low stakes casino. So my question to you is, would
you want to play this game? And don't put the
fun factor into it. Just economically, does
it make sense for you to actually play this game? Well, let's think through the
probabilities a little bit. So, first of all,
what's the probability that the first marble
you pick is green? Actually, let me just
write, first green, probability first green. Well, the total possible
outcomes-- there's 5 marbles here,
all equally likely. So there's 5 possible outcomes. 3 of them satisfy your event
that the first is green. So there's a 3/5 probability
that the first is green. So you have a 3/5 chance,
3/5 probability I should say, that after that first
pick you're kind of still in the game. Now, what we really care
about is your probability of winning the game. You want the first to be
green, and the second green. Well, let's think about
this a little bit. What is the probability that
the first is green-- first, I'll just write g for green--
and the second is green? Now, you might be
tempted to say, oh, well, maybe the second being green
is the same probability. It's 3/5. I can just multiply
3/5 times 3/5, and I'll get 9/25, seems like
a pretty straightforward thing. But the realization
here is what you do with that first green marble. You don't take that first
green marble out, look at it, and put it back in the bag. So when you take
that second pick, the number of green
marbles that are in the bag depends on what you
got on the first pick. Remember, we take
the marble out. If it's a green
marble, whatever marble it is, at whatever
after the first pick, we leave it on the table. We are not replacing it. So there's not any
replacement here. So these are not
independent events. Let me make this
clear, not independent. Or in particular, the second
pick is dependent on the first. Dependent on the first pick. If the first pick
is green, then you don't have 3 green
marbles in a bag of 5. If the first pick
is green, you now have 2 green marbles
in a bag of 4. So the way that we
would refer to this, the probability of both
of these happening, yes, it's definitely
equal to the probability of the first green times-- now,
this is kind of the new idea-- the probability of the second
green given-- this little line right over here, just a
straight up vertical line, just means given-- given that
the first was green. Now, what is the probability
that the second marble is green given that the first
marble was green? Well, we drew the
scenario right over here. If the first marble
is green, there are 4 possible outcomes,
not 5 anymore, and 2 of them satisfy your criteria. So 2 of them satisfy
your criteria. So the probability of the
first marble green being green and the second
marble being green, is going to be the probability
that your first is green, so it's going to be 3/5,
times the probability that the second is green,
given that the first was green. Now you have 1 less
marble in the bag, and we're assuming that
the first pick was green, so you'll only have
2 green marbles left. And so what does this give
us for our total probability? Let's see, 3/5 times 2/4. Well, 2/4 is the
same thing as 1/2. This is going to be
equal to 3/5 times 1/2, which is equal to 3/10. Or we could write that as 0.30,
or we could say there's a 30% chance of picking
2 green marbles, when we are not replacing. So given that, let me ask
you the question again. Would you want to
play this game? Well, if you played this
game many, many, many, many, many times, on average you have
a 30% chance of winning $1. And we haven't covered this yet,
but so your expected value is really going to
be 30% times $1-- this gives you a little
bit of a preview-- which is going to be $0.30
30% chance of winning $1. You would expect on average,
if you were to play this many, many, many times, that playing
the game is going to give you $0.30. Now, would you want to
give someone $0.35 to get, on average, $0.30? No, you would not want
to play this game. Now, one thing I will
let you think about is, would you want to play
this game if you could replace the green marble,
the first pick. After the first pick, if you
could replace the green marble, would you want to play
the game in that scenario?