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So welcome back.

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I think the plan is now clear.

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We know that we can estimate PI by calculating four times the ratio of the RAAF to circle divided by

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the area of the square.

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And this ratio of these two areas we can determine by generating points randomly and just by counting

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how many of them are inside and outside.

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So let's go ahead and do this.

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First of all, we must, of course, define how many points we want to have.

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So I write points is equal to and let's start small.

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Let's just use a thousand points.

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And then I can write Rand is equal to and p dot random thoughts rand.

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And then I write points.

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So maybe make this number even smaller.

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Let's just use ten.

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And so if I now call and we have an array of 10 different numbers in the range of zero and one.

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In this case, it's a bit odd because we have three of these points that are very close to one.

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But if I repeat this, then you see it changes.

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So really, these are randomly generated numbers from what we want to have here.

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When we look at our graph is we want to have points in the range of minus one and one.

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So I write two times this random number minus one.

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This time we have points in the range of minus one and plus one.

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Let's do it again c from minus one to plus one.

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This one here is pretty close to +1.

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So you see it works.

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Now the thing is here we just have 10 points, but just basically in one dimension.

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So just a single coordinates and we want to have X and Y coordinates.

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So we will generate here a number of 20 points in our case of two times points.

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So then we have 20 of them and then we will write random three shape points.

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Comma to what this does is it generates pairs of these numbers.

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So basically first we generate 20 points and then we will order them.

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We'll just take the first to make them a list.

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Second, to make them in the list.

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And so now we have these pairs of these coordinates and these will be 10 of our points.

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OK, so let's cut this, we don't need it anymore.

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And this one we can call friend points.

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OK.

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And as I said previously, we need, of course, more points.

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Let's start with a thousand.

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So now we have hour variable rent points, 1000 points with two coordinates each, and each of these

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coordinates is in the range of minus one and +1.

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Now we need the norm.

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So basically, we need the criterion to decide whether or not the points inside or outside the circle.

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So if we take the norm, so basically the distance of the points to the center, then we know if it

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is outside or inside.

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Because if this length, this distance is smaller than the radius, then it must be inside.

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And if it is larger and the radius, then it must be outside.

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So we need to know and this will be the criterion.

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So I write norm points and to calculate this, we basically take our list of red points and we take

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every single one of them.

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And for every single one of these points, we take the first coordinates, square it and we take the

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second coordinates and square and at these two values.

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So now we could right here NPR dot square root to have the actual norm.

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But to be honest, it makes the code a bit slower and it's not really required, so we can just leave

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it out here.

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So this will now actually be the norm square points list.

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But I just call it norm points.

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Even though this is not the actual norm, this is the square of the norm.

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However, the good thing is when the radius is one and also the square radius will be one.

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So we can now just see it say if the square of the norm is smaller than one, then it's inside.

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And if the square of the norm is larger than one, then it's outside.

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So there's value here.

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This list of values will contain valuable information to determine this ratio here.

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And so what we have to do next is we have to distinguish these points based on this criterion that I

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just mentioned.

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So, all right, points out, is equal to random points.

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So Rand's points recorded it.

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So it will be this list, but only the elements which fulfil a certain criterion.

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And this is pretty easy to implement in Python.

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It just writes the criterion that you want to use, which in this case is also a list of the same length

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and larger than one.

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So what happens here is the following.

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So it will start with the first entry in the round points and then it will determine based on the first

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entry of this list.

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So it will see OK, round points.

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First element will be this one norm points.

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The first element will be

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what's happening.

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No points.

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Oh, I didn't try it.

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OK, so now Norm points, the first element is this one, so the length of the of this off the distance

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of this point compared to the center, Kornet squared is equal to zero point four two.

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So there's is smaller than one.

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So it will not be in this case, so it will not be in points out.

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Instead, it will be in points in where the condition will be like this.

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So it doesn't really matter if you use your smaller and equal or two smaller because due to the random

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function, the chance is practically zero that there is a point where the norm is exactly equal to one.

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So it doesn't really matter if we use this or this.

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So like this, it should work.

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And now if we look at brands, if we look at points out the first element we see again our point from

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before.

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No, we don't.

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Of course, no, we don't, because it's not in here, but it's instead and points in.

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So here we see it.

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Yeah, here it is.

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It's the same point.

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And you see or distinction worked.

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So now this point is only and points in and it's not and points out.

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So no, I will delete all of these test cases here that I've just added to show you what is happening

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here.

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And these are the commands that we need.

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Rand runs, points, norm points, points out, points in.

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And to be honest, we only really need the points in.

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So we could also comment this one because this is not really a requirement.

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This is because we know already how many points we have, and we only need to know how many of them

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are in site.

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OK.

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So to get the result, we are almost finished now, we can just write pipe approx.

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Is equal to four times the ratio of the area circle divided by area of square.

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So it would be the number of points inside divided by the total number of points.

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So it would be the length of points in divided by the total number of points and to plot the or two

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to output the result.

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We just write print pi approx.

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And you see it's three point one two four.

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So we can compare this with the actual value.

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And you see the difference is not that big.

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So quite a good result.

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Of course, we can now go ahead and just increase this number.

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And the result got a bit better.

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Christie's number even more, and the result got even better.

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So of course, if we just repeat the same thing, then the value can change a bit because it is really

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given by random.

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So to show you or to visualize this a bit more, let's make a nice plot.

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Let's take the plot from before, and let's change it a bit.

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That's just add here these points that we have generated, and I will do this in the following way.

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I would just add here to scatter plots, so I will write points in.

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And we use all the points and the X coordinates, and for the Y coordinates, we ride like this.

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Then I use color for the inside points to be,

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I don't know, green green for positive and then the size of the points we can also specify, so make

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the points a bit smaller.

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So here we don't see anything does this because these patches are executed last, so they are intel

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on top of the points.

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And actually, we don't really need the rectangle anymore.

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So here you see all the points that are inside, so you see it worked.

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And then we can also add the points that are outside.

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So I will remove this comment here.

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So you see, we we only need the list points outside for plotting what not really to determine the value

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of PI.

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So, all right, points out comes out and the color will be red.

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And you see like this, we have now a nice plot of all of the points.

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So we have now how many have been 100000 points?

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We generate 100000 points with two coordinates each and we have put it inside the square and there are

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some of them are inside the circle and some of them are not inside the circle.

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And just by comparing the ratio of the of the of the number of points, we have determined the value

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of PI and we determined this value pretty accurately.

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So this is a method that works very, very well.

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It's based on lists and based on a race.

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So you see, we have first generated this parade and we've reshaped it.

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Then we've generated the second array with the norm than we have generated a new array where we have

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just taken some of the values from the old array based on this condition.

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And then we have just calculated the length.

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So maybe if you are not so familiar with these arrays and all of these commands, this looks a bit difficult

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to you.

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So it is sometimes even easier to just use a loop methods and it would, I think, even be less lines

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of code to do this.

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And as we will do in the next lecture, what I can tell you already, this is such a loop method is

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much, much slower, but it's sometimes easier.

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And I want to show it to you anyway.