Introduction to dependent probability Deciding whether you want to play a game at a strange casino.
Introduction to dependent probability
- 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?
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