1 00:00:00,060 --> 00:00:02,850 So let's get started with a very famous example. 2 00:00:02,940 --> 00:00:10,380 We're going to calculate the value of PI using a Monte Carlo algorithm or better to say we're going 3 00:00:10,380 --> 00:00:12,810 to approximate the value of PI. 4 00:00:13,440 --> 00:00:19,920 And we will exploit randomness, randomness and generated numbers to determine this value. 5 00:00:20,370 --> 00:00:23,820 So if you have never seen this, I think this is a pretty impressive example. 6 00:00:23,820 --> 00:00:28,590 But if you have seen this and I think it's pretty clear what's going to happen because this is a pretty 7 00:00:28,590 --> 00:00:29,700 standard example. 8 00:00:30,840 --> 00:00:35,880 So first of all, we will note NUM Pi and model partnership PI plot as before. 9 00:00:36,180 --> 00:00:43,350 And now we will also import another module from Protip, which is called patches, and I write down 10 00:00:43,830 --> 00:00:45,540 dot patches as patches. 11 00:00:46,500 --> 00:00:51,870 So let's run this and now we can do is we can basically draw geometric figures. 12 00:00:51,900 --> 00:00:55,260 So I want to draw here square and also a circle. 13 00:00:56,400 --> 00:00:59,250 So let's go ahead and I'll show you how this works. 14 00:01:00,210 --> 00:01:04,080 So we can do this as follows right rectangle. 15 00:01:05,280 --> 00:01:09,510 So this is where it this will be where the rectangle is stored and then they are just right patches 16 00:01:10,260 --> 00:01:13,530 dot capital R Act angle. 17 00:01:15,120 --> 00:01:19,590 And then we must define some values for the rectangle. 18 00:01:20,040 --> 00:01:23,100 So first of all, we must find the bottom left corner. 19 00:01:23,550 --> 00:01:27,210 So this will be the coordinates minus one and minus one in my case. 20 00:01:27,810 --> 00:01:33,030 Then we must define the width and the height, which is two and two. 21 00:01:33,390 --> 00:01:39,780 So basically, now it goes from minus one to plus one in the X and then the Y --. 22 00:01:40,770 --> 00:01:43,210 And then we can define some color. 23 00:01:43,320 --> 00:01:48,290 For example, I write phase color equal to reds. 24 00:01:49,740 --> 00:01:51,330 So if I run this, nothing happens. 25 00:01:51,330 --> 00:01:56,910 We must first add this to a plot for this, I write. 26 00:01:58,350 --> 00:02:02,820 Figure is equal to P.L. T dot figure. 27 00:02:03,280 --> 00:02:06,180 It's basically here we are initializing to figure. 28 00:02:06,540 --> 00:02:09,990 And then I think we did this before a several times. 29 00:02:09,990 --> 00:02:13,820 I will just try to figure out, add subplots. 30 00:02:14,040 --> 00:02:15,900 So this is a bit tricky. 31 00:02:15,900 --> 00:02:20,160 I think about Python the way it has to be so difficult for plotting. 32 00:02:20,520 --> 00:02:24,540 But this is the same thing that we had to do when we wanted to change the aspect ratio. 33 00:02:25,170 --> 00:02:31,620 So here we can all rights aspects equal to equal. 34 00:02:33,240 --> 00:02:39,870 And now we can write X thoughts at underscore patch. 35 00:02:40,860 --> 00:02:44,640 Now we write what we want to add and we want to add a rectangle. 36 00:02:44,700 --> 00:02:50,970 The thing that we have to find previously and now we have two plots with a red rectangle. 37 00:02:52,530 --> 00:02:58,860 So to show you that this is really just a rectangle, I can also specify the limits. 38 00:02:59,370 --> 00:03:01,440 So please dot x limb. 39 00:03:02,850 --> 00:03:09,990 For example, we could write minus one point one and one point one and the same for Y. 40 00:03:12,960 --> 00:03:17,490 So then you see the red rectangle really goes just from minus one to one. 41 00:03:17,910 --> 00:03:19,290 Two looks pretty nice, I think. 42 00:03:19,920 --> 00:03:22,570 And now I will add a circle. 43 00:03:23,730 --> 00:03:26,340 So circle equals patches. 44 00:03:26,790 --> 00:03:29,250 Dots circle here. 45 00:03:29,250 --> 00:03:30,030 The optional. 46 00:03:30,340 --> 00:03:31,400 No, they are not optional. 47 00:03:31,410 --> 00:03:32,860 They are required arguments. 48 00:03:32,910 --> 00:03:35,580 They are at the center coordinate position. 49 00:03:35,590 --> 00:03:41,250 So 0.0 in our case and then the radius and then for the face color. 50 00:03:42,630 --> 00:03:43,380 All right. 51 00:03:43,920 --> 00:03:44,370 None. 52 00:03:44,880 --> 00:03:46,290 So then it's transparent. 53 00:03:47,850 --> 00:03:49,890 And for edge color. 54 00:03:51,240 --> 00:03:53,220 All right, blue. 55 00:03:56,450 --> 00:04:02,180 And when I ran this, nothing happens because first person must have this two hour plot. 56 00:04:02,750 --> 00:04:07,520 So we must write x dot and underscore patches circle. 57 00:04:08,840 --> 00:04:13,640 And now you see, we have now there's redirect angle and the blue circle. 58 00:04:14,990 --> 00:04:18,709 So now let me show you what I wrote down previously by deleting these cells. 59 00:04:19,100 --> 00:04:21,890 So the whole trick about this? 60 00:04:22,670 --> 00:04:31,190 So, yeah, the whole trick is that we consider this as a sketch for what is going to happen next. 61 00:04:32,360 --> 00:04:37,890 We will generate randomly some points in this red square. 62 00:04:38,390 --> 00:04:44,960 So just randomly, we pick some points in here and then we count how many points are inside and how 63 00:04:44,960 --> 00:04:47,630 many points are outside of the circle. 64 00:04:48,770 --> 00:04:53,720 And by doing this, we can determine the ratio of these two areas. 65 00:04:54,560 --> 00:05:01,190 So for example, if we have 1000 points and 700 of them would be inside the circle and 300 would be 66 00:05:01,190 --> 00:05:08,540 outside, then the ratio of the circle to the square would be 700 over 1000. 67 00:05:08,540 --> 00:05:10,340 So zero point seven. 68 00:05:11,360 --> 00:05:14,180 However, this is not the actual value we would get. 69 00:05:14,840 --> 00:05:16,130 So we will see this in the second. 70 00:05:17,030 --> 00:05:22,640 But the thing is, we know already from mathematics what is the area of the circle and what is the area 71 00:05:22,640 --> 00:05:23,390 of the square? 72 00:05:24,110 --> 00:05:27,170 So of course, from the circle, it's Pi Times Square. 73 00:05:27,410 --> 00:05:34,910 In our case, the radius is one, so the area would just be PI and the area of the square would be this 74 00:05:34,910 --> 00:05:36,260 length, he squared. 75 00:05:36,740 --> 00:05:39,380 And this length is two and two squared is four. 76 00:05:40,940 --> 00:05:48,440 So we know that the ratio of a circle divided by a square is PI over four. 77 00:05:49,730 --> 00:05:56,210 And so we can solve this for PI and we will get PI is equal to four times the ratio. 78 00:05:57,320 --> 00:06:02,270 And the thing is that as I have just explained, by generating these random numbers and by counting 79 00:06:02,270 --> 00:06:09,140 how many of them are inside and outside of the circle, we can determine this ratio just numerically 80 00:06:09,500 --> 00:06:13,070 just by generating a large number of points and just by counting them. 81 00:06:14,150 --> 00:06:20,180 And so when we multiply this ratio by four, then we get the approximate value of PI. 82 00:06:21,050 --> 00:06:25,040 And the next lecture, I want to show you how this works and how you can implement this.