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Studying for a test? Prepare with these 4 lessons on Probability.
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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?