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PROFESSOR: So, if you remember, just before the break, as long ago as it was, we had looked at the problem of fitting curves to data. And the example we had seen, is that it's often possible, in fact, usually possible, to find a good fit to old values. What we looked at was, we looked at a small number of points, we took a high degree polynomial, sure enough, we got a great fit. The difficulty was, a great fit to old values does not necessarily imply a good fit to new values. And in general, that's somewhat worrisome.

So now I want to spend a little bit of time I'm looking at some tools, that we can use to better understand the notion of, when we have a bunch of points, what do they look like? How does the variation work? This gets back to a concept that we've used a number of times, which is a notion of a distribution. Remember, the whole logic behind our idea of using simulation, or polling, or any kind of statistical technique, is the assumption that the values we would draw were representative of the values of the larger population. We're sampling some subset of the population, and we're assuming that that sample is representative of the greater population. We talked about several different issues related to that.

I now want to look at that a little bit more formally. And we'll start with the very old problem of rolling dice. I presume you've all seen what a pair of dice look like, right? They've got the numbers 1 through 6 on them, you roll them and something comes up. If you haven't seen it, if you look at the very back, at the back page of the handout today, you'll see a picture of a very old die. Some time from the fourth to the second century BC. Looks remarkably like a modern dice, except it's not made out of plastic, it's made out of bones. And in fact, if you were interested in the history of gambling, or if you happen to play with dice, people do call them bones. And that just dates back to the fact that the original ones were made that way. And in fact, what we'll see is, that in the history of probability and statistics, an awful lot of the math that we take for granted today, came from people's attempts to understand various games of chance.

So, let's look at it. So we'll look at this program. You should have this in the front of the handout. So I'm going to start with a fair dice. That is to say, when you roll it, it's equally probable that you get 1, 2, 3, 4, 5, or 6. And I'm going to throw a pair. You can see it's very simple. I'll take d 1, first die is random dot choice from vals 1. d 2 will be random dot choice from vals 2. So I'm going to pass it in two sets of possible values, and randomly choose one or the other, and then return them. And the way I'll conduct a trial is, I'll take some number of throws, and two different kinds of dice. Throws will be the empty set, actually, yeah. And then I'll just do it. For i in range number of throws, d 1, d 2 is equal to throw a pair, and then I'll append it, and then I'll return it. Very simple, right? Could hardly imagine a simpler little program.

And then, we'll analyze it. And we're going to analyze it. Well, first let's analyze it one way, and then we'll look at something slightly different.

PROFESSOR: So, if you remember, just before the break, as long ago as it was, we had looked at the problem of fitting curves to data. And the example we had seen, is that it's often possible, in fact, usually possible, to find a good fit to old values. What we looked at was, we looked at a small number of points, we took a high degree polynomial, sure enough, we got a great fit. The difficulty was, a great fit to old values does not necessarily imply a good fit to new values. And in general, that's somewhat worrisome.

So now I want to spend a little bit of time I'm looking at some tools, that we can use to better understand the notion of, when we have a bunch of points, what do they look like? How does the variation work? This gets back to a concept that we've used a number of times, which is a notion of a distribution. Remember, the whole logic behind our idea of using simulation, or polling, or any kind of statistical technique, is the assumption that the values we would draw were representative of the values of the larger population. We're sampling some subset of the population, and we're assuming that that sample is representative of the greater population. We talked about several different issues related to that.

I now want to look at that a little bit more formally. And we'll start with the very old problem of rolling dice. I presume you've all seen what a pair of dice look like, right? They've got the numbers 1 through 6 on them, you roll them and something comes up. If you haven't seen it, if you look at the very back, at the back page of the handout today, you'll see a picture of a very old die. Some time from the fourth to the second century BC. Looks remarkably like a modern dice, except it's not made out of plastic, it's made out of bones. And in fact, if you were interested in the history of gambling, or if you happen to play with dice, people do call them bones. And that just dates back to the fact that the original ones were made that way. And in fact, what we'll see is, that in the history of probability and statistics, an awful lot of the math that we take for granted today, came from people's attempts to understand various games of chance.

So, let's look at it. So we'll look at this program. You should have this in the front of the handout. So I'm going to start with a fair dice. That is to say, when you roll it, it's equally probable that you get 1, 2, 3, 4, 5, or 6. And I'm going to throw a pair. You can see it's very simple. I'll take d 1, first die is random dot choice from vals 1. d 2 will be random dot choice from vals 2. So I'm going to pass it in two sets of possible values, and randomly choose one or the other, and then return them. And the way I'll conduct a trial is, I'll take some number of throws, and two different kinds of dice. Throws will be the empty set, actually, yeah. And then I'll just do it. For i in range number of throws, d 1, d 2 is equal to throw a pair, and then I'll append it, and then I'll return it. Very simple, right? Could hardly imagine a simpler little program.

And then, we'll analyze it. And we're going to analyze it. Well, first let's analyze it one way, and then we'll look at something slightly different.

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