Dependent probability introduction

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 three green marbles - orange marbles and I'm going to stick them in the bag and he literally sticks them into the empty bag to show you there's truly three green marbles three green marbles and two orange marbles - orange marbles and he says the game that I want you to play or that if you 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 you're going to win the prize of you're going to win win $1 if you get 2 greens if you get 2 greens you're going to win $1 we said well this sounds like an interesting game how much does it cost to play and the guy tells you it is 35 cents 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 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 what's the probability that first marble first marble is green actually let me just write first green probability first green first green well the total possible outcomes there's five marbles here all equally likely so there's five possible outcomes three 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 thirty fifth chance 3/5 probability I should say that on that after that [ __ ] you're kind of still in the game now what's the problem that we 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 and the second 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 over 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 that 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 not or in particular the second pick the second pick is dependent on the first dependent dependent on the first on the first pick if the first pick is green then you don't have three green marbles in a bag of 5 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 the probability of both of these happening yes it's definitely equal to the probability of the first green for the probability of the first green times times now this is kind of the new idea the probability of the second green second green given this little line right over here just a 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 draw through the 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 green being green and the second marble 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 one less marble in the bag and we're assuming that the first pick was green so you only have two marble to green marbles left and so what does this give us for our total probability well let's see 3/5 times 2 4 it's 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 tenths or we could write that as zero point three zero or we could say there's a 30% 30% 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 many times on average you have a 30% chance 30% chance of winning 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 30 cents 30 cents 30% chance of winning $1 you would expect on average if you 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 some 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 pick would you want to play the game in that scenario