Let's imagine ourselves in some kind of 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,
three green marbles, two orange marbles, and I'm gonna stick them in the bag
And he literally sticks them into the empty bag
To show you that there is truly three green marbles, and two 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 two green marbles
If you're able to take one 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 of
You're going to win one dollar if you get two greens.
Well you say, "this sounds like an interesting game,
How much does it cost to play?"
And the guy tells you it is 35 cents to play.
So obviously, fairly low stakes casino.
So my question to you is, would you want to play this game?
And don't put, you know, the fun factor into it
Just economically, does it makes 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?
What's the probability that first marble is green?
Actually, just let me 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 three-fifths probability that the first is green.
So you have a three-fifths chance
Three-fifths 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
I'll just write "g" for green
And the second is green.
Now, you might be tempted to say
"Oh well the second being green is the same probability,
it's three-fifths. I can just multiply three-fifths times three-fifths
And I'll get nine over twenty-five
Seems like a pretty straight-forward 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 or whatever marble it is
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 three green marbles in a bag of five
If the first pick is green, you now have two green marbles in a bag of four
So the way that we would refer to this is 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 over here
just this straight up, vertical line just means given
Given, this 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 this scenario right over here
If the first marble is green there are four possible outcomes
not five anymore
And two of them satisfy your criteria.
So two of them satisfy your criteria.
So the probability of the first marble 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 three-fifths
Times the probability that the second is green given the first was green.
Now you have one less marble in the bag and we're assuming that the first pick was green
So you only have two green marbles left.
And so what does this give us for our total probability?
Let's see. Three-fifths times two-fourths
well two-fourths is the same thing as one half
This is going to be equal to three-fifths times one half
Which is equal to three tenths
Or we could write that as zero point three zero
Or we could say that there is a 30 percent chance
of picking two 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 times
On average, you have a 30 percent chance
of winning one dollar.
And we haven't covered this yet,
So your expected value is really going to be
30 percent times one dollar
This gives you a little bit of a preview
Which is going to be thirty cents
Thirty percent chance of winning one dollar
You would expect, on average,
if you were to play this many, many, many times
that playing the game is going to give you 30 cents.
Now, would you want to give someone
35 cents to get on average 30 cents?
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?