Introduction to dependent probability

Deciding whether you want to play a game at a strange casino.

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?