WEBVTT 1 00:00:01.794 --> 00:00:10.494 Sorry about that I'll review the 1st couple of minutes that no, that is weird. My primary laptop doesn't even give recording as an option. 2 00:00:10.494 --> 00:00:18.954 My secondary laptop that I'm using to see what you see did have recording as an option go figure. Okay. So. 3 00:00:20.640 --> 00:00:26.399 So, let me just so it's for people watching the recording later. 4 00:00:26.399 --> 00:00:37.170 People different time zones. Okay so today we'll finish off chapter 2 get into chapter 3. I mentioned. 5 00:00:37.170 --> 00:00:41.399 Serious of recording now. 6 00:00:43.229 --> 00:00:47.490 Yeah, okay. 7 00:00:48.960 --> 00:00:57.329 Ok, so exam next Monday, we'll use grade scope. We'll have a 2nd. 8 00:00:57.329 --> 00:01:01.679 Administration like, 12 hours later. 9 00:01:01.679 --> 00:01:11.310 Number questions depends on how big each question is There'll be designed to be easy to grade with short answers based on what I've covered in. 10 00:01:11.310 --> 00:01:24.204 Class and the homeworks, so I will not ask questions based directly on what Professor Rad Keith did in his videos there for enrichment stuff unless I had a question on that. 11 00:01:24.204 --> 00:01:27.685 And which forced you and then we may be work that question. 12 00:01:27.959 --> 00:01:33.090 The goal is that you not be under time pressure. 13 00:01:33.090 --> 00:01:37.560 We may not succeed that, but that's the goal. 14 00:01:38.640 --> 00:01:47.040 Um, okay, and then the other thing I mentioned is an enrichment thing on here in the real world thing you can click on. Okay. 15 00:01:47.040 --> 00:01:52.739 So, new material now is geometric probability law. 16 00:01:52.739 --> 00:02:07.680 Give you an example over here, let's say we'll drill for oil probability, fitting oil. When you drill a hole is say 1001 and 100 I said, and the random variable K is the. 17 00:02:07.680 --> 00:02:17.069 Number of holes the 1st success. 18 00:02:17.069 --> 00:02:22.500 Okay, and the proper bells here say K, girls. 19 00:02:22.500 --> 00:02:28.379 On the 1st hole is point 0, 1 probability of happening on the 2nd whole. 20 00:02:28.379 --> 00:02:35.400 Well, it's going to be 99% chance. You failed on the 1st, 1% chance who succeeded on the 2nd. 21 00:02:35.400 --> 00:02:43.169 Probably some general K would be 99 to the K minus 1 times. 2.01. 22 00:02:43.169 --> 00:02:46.469 And so, okay, examples here. 23 00:02:46.469 --> 00:02:51.659 And now we can get the probability that. 24 00:02:51.659 --> 00:02:55.650 More than Kate trials are required. 25 00:02:55.650 --> 00:03:00.330 It is a big 1, so to hear. Okay. 26 00:03:00.330 --> 00:03:07.020 And here, the probability of a success at any given 1 is point 0 1. so that's. 27 00:03:07.020 --> 00:03:11.849 Be using the notation of here now. 28 00:03:11.849 --> 00:03:15.780 Probability that more than K trials. 29 00:03:15.780 --> 00:03:18.960 Are required. 30 00:03:26.189 --> 00:03:29.580 And K, trials were required. 31 00:03:31.349 --> 00:03:34.860 That's going to be the. 32 00:03:34.860 --> 00:03:43.710 of i will say k plus one to infinity of . 33 00:03:45.689 --> 00:03:51.810 1, minus P2 the minus 1 times. 34 00:03:51.810 --> 00:04:00.150 P, so so is the number so, this would be the 1st success on the ice trial for I, from K plus 1 up. 35 00:04:02.310 --> 00:04:07.289 Combine here and. 36 00:04:07.289 --> 00:04:12.449 So that's going to be, I'll rewrite it eventually. That will scroll up. 37 00:04:12.449 --> 00:04:18.870 Remote monitoring thing is not scrolling. There we go. So, that was the sum of all. 38 00:04:21.839 --> 00:04:32.338 And we can pull things out. 39 00:04:47.699 --> 00:04:51.269 Oh, we can pull out the P. well, we can also do. 40 00:04:51.269 --> 00:05:00.869 Is I'm no pull out the P and then that would be instead of I minus 1. this would be 1 minus P2 the. 41 00:05:02.668 --> 00:05:16.259 actually i say zero to infinity and this could be . 42 00:05:17.548 --> 00:05:27.718 Is this correct? I may have a minus 1 here. The initial thing here is K, the initial thing here is K plus 100 K. 43 00:05:29.999 --> 00:05:40.769 Okay. 44 00:05:40.769 --> 00:05:45.718 And this some here, that's a geometric series. 45 00:05:45.718 --> 00:05:55.468 And then some of that display little aside here, put in different color, let's say red here. If you have the sum of all the. 46 00:05:57.389 --> 00:06:05.519 Okay. 47 00:06:05.519 --> 00:06:10.619 Correct me? If I make too many mistakes. 48 00:06:10.619 --> 00:06:14.579 So so that thing up there. 49 00:06:14.579 --> 00:06:22.199 He calls stop it. 50 00:06:22.199 --> 00:06:28.468 Okay. 51 00:06:30.269 --> 00:06:36.119 And this thing up here is 1 over, which counsels and. 52 00:06:38.338 --> 00:06:43.288 Okay, which is what the book had. Okay. 53 00:06:43.288 --> 00:06:50.488 Let's put a box around this, so, in case, if somebody's looking at this in black and white, and then this here. 54 00:06:50.488 --> 00:06:58.108 Down there. Oh, okay. And gave lots of examples here. And so on. 55 00:07:00.834 --> 00:07:11.394 And we'll see another example later. So, next topic is sequences of dependent experiments and they're called markup change. So, this is not very important in communication. 56 00:07:11.394 --> 00:07:21.324 For example, if you, for example, have English text, a Q, is almost always followed by you. There are some counter examples. 57 00:07:21.749 --> 00:07:26.668 Country name or act, for example, but it's not very common. 58 00:07:26.668 --> 00:07:31.319 And so if we're looking at probability of letters in English. 59 00:07:31.319 --> 00:07:35.788 You know, the previous letter was accused that probably at this letter as a, you is. 60 00:07:35.788 --> 00:07:38.999 Not far short of 100% for example. 61 00:07:40.944 --> 00:07:44.153 Give other examples here and this is useful for example, 62 00:07:44.153 --> 00:07:45.413 if you're doing compression, 63 00:07:45.653 --> 00:07:51.533 if you want to compress the text to take the smallest amount of space or if you're doing crypt analysis, 64 00:07:51.533 --> 00:07:55.134 this is also you do the opposite is also analogy, 65 00:07:55.283 --> 00:07:56.363 speech recognition, 66 00:07:56.363 --> 00:07:58.103 phone names and that sort of thing, 67 00:07:58.824 --> 00:08:00.144 and in fact. 68 00:08:00.988 --> 00:08:14.189 Dang, this, in fact, is, I think the secret to a machine understanding of speech speech recognition is not for a long time. They tried using sophisticated models of language. 69 00:08:14.189 --> 00:08:25.139 And they didn't work, so they just went now that uses brute force using simple correlation, cable, simple mark off change and past computers. 70 00:08:25.139 --> 00:08:28.199 So, good for sometimes works. 71 00:08:28.199 --> 00:08:32.009 I'm looking at an example or 2. 72 00:08:32.009 --> 00:08:37.708 This is my textbook here. 73 00:08:40.379 --> 00:08:46.139 Oops, can stop. 74 00:08:47.308 --> 00:08:54.899 Air controlled by re transmission, for example. 75 00:08:54.899 --> 00:09:08.339 And this would be, this would be the geometric thing. Don't just like drilling for oil and to succeed. This would be transmitting until you succeed. The math is the same. 76 00:09:08.339 --> 00:09:11.788 24344. 77 00:09:11.788 --> 00:09:17.369 Example that is. 78 00:09:18.688 --> 00:09:21.688 Or repeatedly drawing a ball. 79 00:09:21.688 --> 00:09:25.168 So, I'll go to 245. that's. 80 00:09:25.168 --> 00:09:30.688 Not so interesting that we can look at a secret. What they do is that. 81 00:09:30.688 --> 00:09:37.589 They well, let me look at that. That's a nice example of it. 82 00:09:37.589 --> 00:09:43.198 Okay, stop so okay. We've got 2 earns which balls in them. 83 00:09:43.198 --> 00:09:54.328 And so the earned 0T and earned 1 are 0T has 2 0T balls and a 1 ball. 84 00:09:54.328 --> 00:09:58.139 1 has 5, 1 balls. 85 00:09:58.139 --> 00:10:02.219 And a 0T ball and. 86 00:10:02.219 --> 00:10:06.989 And basically what we do. 87 00:10:06.989 --> 00:10:19.528 Is that we keep drawing balls until something happens until we get 2 balls on the same type or something? And the way to look at that is some experiment like. 88 00:10:21.899 --> 00:10:27.149 We start off here and then we toss a coin heads or tails. 89 00:10:27.149 --> 00:10:33.479 Heads we go to our 0T tales we go to her 1 and then we draw. 90 00:10:34.678 --> 00:10:48.234 And that, and then we draw a ball at random and we follow the arrow depending on which 1. so, these are the states that can happen. I'm just giving an executive summary of this. And then what we can do is we can attach probabilities to this. 91 00:10:48.234 --> 00:10:48.413 So, 92 00:10:48.413 --> 00:10:49.943 we started to start 50, 93 00:10:49.943 --> 00:10:50.994 50 we go to 0, 94 00:10:51.023 --> 00:10:55.464 1 and then if we're up here where where I got the purse, 95 00:10:55.464 --> 00:11:04.734 or you can see that we're in 0T and which had the 2 0T balls and the 1 1 ball we draw a ball at random. 96 00:11:05.009 --> 00:11:12.928 Then we see a 0T or we see a 1 and so we've attach probabilities. So these are transition probabilities. 97 00:11:12.928 --> 00:11:17.489 If we're in this state, the probability that we would transition to 1 of those to. 98 00:11:17.903 --> 00:11:32.453 If where in earn 1, then it's 5 to 1 6, 6, 6 balls. 1 of them is a 0, 5 of them are 1 so that we have probabilities on going to the next state here. 99 00:11:32.813 --> 00:11:33.474 And then. 100 00:11:33.958 --> 00:11:39.178 We have all these transition probabilities and this is what a mark off chain looks like here. 101 00:11:39.833 --> 00:11:50.274 Now, anticipating a little if you want, you can write to transitions actually, as a matrix, and you can have a vector, which is a probability here in either state. 102 00:11:50.663 --> 00:12:01.974 And then you have a 2 by 2 a matrix, which would be transit, which will then you'd modified by this probability factor and give you the probability vector that gear in another state that the next state and you can find. 103 00:12:02.249 --> 00:12:06.778 If the thing settles out in some state probability and stuff like that. 104 00:12:06.778 --> 00:12:14.038 Now, this example, a little confusingly written, because for the 1st step that they label. 105 00:12:14.038 --> 00:12:22.168 Urine 0T, I want to label the balls there on 1 also and I think the reason they do that is they can make this diagram look more simple. 106 00:12:22.168 --> 00:12:33.479 So, here in the 1st, well, you mentioned the 1st state, you haven't done anything the next state you've picked earn waters is there or 1 or the next state is we see a ball 0T or 1. 107 00:12:33.479 --> 00:12:38.999 So, here the 0T means are and we're doing our in 0T here. It means we. 108 00:12:38.999 --> 00:12:45.359 See, as arrow. Sorry so little confusing, but that's the. 109 00:12:45.359 --> 00:12:57.808 Other quick question yes, probably that get over the beginning state would be 1 half times a 3rd plus 1 half times. 5. 6 correct? 110 00:12:57.808 --> 00:13:01.259 Okay, yeah, for the rest of you. 111 00:13:01.259 --> 00:13:13.583 That sequence dependent thing so the probability that you would see a 1 ball, while there's 2 ways you could do it, you could see the 1 ball in earn 0T, or you could see the 1 ball in earned 1. 112 00:13:14.124 --> 00:13:19.374 so, it's a total probability thing. I would anyone like me to write it out or. 113 00:13:19.619 --> 00:13:26.009 Speak up. Okay. Okay. 114 00:13:26.009 --> 00:13:35.639 Same sort of idea here. In fact, they're sort of what you're asking about the probability that you see him. Okay. 115 00:13:36.264 --> 00:13:48.083 That's not the best example, because of the confusion or something, but that's the thing is, they've got the sequences of dependent experiments and the probabilities change as you move along them and great practical importance. 116 00:13:48.264 --> 00:13:52.374 It's also the secret behind a lot a lot of machine learning and. 117 00:13:52.918 --> 00:13:59.399 And so on, you've got States and transitions I'm not an expert in that, but okay now. 118 00:13:59.399 --> 00:14:05.369 2, 7 here, this is enrichment stuff. Let me see what I wrote down here. 119 00:14:05.369 --> 00:14:10.619 Um, I had here, so the mark off changed the sequences of experiments and so on. 120 00:14:11.759 --> 00:14:16.048 Mention the real world thing, I'll do some review things. 121 00:14:16.048 --> 00:14:23.548 A few of these things to help review it, but I just want to summarize this thing here quickly. 122 00:14:23.548 --> 00:14:27.298 Add a blurb on it. I had a class or 2 ago. 123 00:14:27.714 --> 00:14:35.124 Executive summary don't try to do this on your own if you can you want to use packages there's 2 types of random numbers. 124 00:14:35.153 --> 00:14:49.793 There is the genuine random numbers, which you use some sort of environmental noise, radioactive decay, electronic noise or something and you somehow measure that. And that gives you a random bit perhaps you have to flip flop randomly. 125 00:14:50.303 --> 00:14:53.634 And so every time you do this, you'll get a different number. And this would be the way. 126 00:14:53.969 --> 00:15:04.499 Well, if you're in Linux, for example, slash slash random is using environmental noise stuff in the operating system to try and give you genuine random bits and you would use those. 127 00:15:04.499 --> 00:15:07.979 For example, to give you the key for a cipher. 128 00:15:07.979 --> 00:15:11.249 And it's. 129 00:15:11.249 --> 00:15:19.708 Even this can be tricky to do. Right? I mentioned the book, the 1M, random digits before, and they, the. 130 00:15:19.708 --> 00:15:25.589 The introduction to the book talks about how they did it, they use some electronic noise. They. 131 00:15:25.589 --> 00:15:39.089 Have some sort of something flipping and flopping for a while and they stop and look as it's there were 1. the trouble is that the plot was not completely symmetric. It might spend more time on those arrows and on the ones and so on. It'd be difficult to get a completely symmetric. 132 00:15:39.089 --> 00:15:44.188 Given that discrete electronic devices. Their precision is something like. 133 00:15:44.188 --> 00:15:53.729 20% or whatever, and not even a whole significant digit often. So, what they did is they would have pairs of numbers, and it would add them up. And that would be. 134 00:15:53.729 --> 00:15:57.058 The source of or 2 bits would actually. 135 00:15:57.058 --> 00:16:10.678 Even if either bed was a 1 more than a 0T more than half the time, if you do take 2 bits into the exclusive or if they're not correlated would be better. But so that's hard. Those are genuine random numbers. The. 136 00:16:10.678 --> 00:16:15.719 Often you want to use pseudo random numbers, which are generated by some. 137 00:16:15.719 --> 00:16:23.099 Algorithm in the computer, and the thing was the pseudo random number it says, every time you start. 138 00:16:23.099 --> 00:16:28.708 Your program you get the same sequence of numbers now you might think that it's bad. 139 00:16:28.708 --> 00:16:40.948 But it can be good because every time you run the program, you get the same answer and so it's easier to duplicate the program. If you're doing some sort of experiments, things are reproducible. 140 00:16:40.948 --> 00:16:50.458 And so, if you're around, if you should have run, remember January is really good and sit around the numbers are fine. And if you want them to be really random, you can see the thing. You can start the thing off the genuine random number. 141 00:16:50.458 --> 00:16:59.519 And so, on the pseudo random numbers here, they talk about that here. There's something called a twist or which is a good 1 to use. So. 142 00:16:59.519 --> 00:17:06.989 There's old things called linear congruent show, random number of generators that turned out to be bad, but they were used for many years. 143 00:17:06.989 --> 00:17:12.118 And you got packages and so on, which will do it for you, Matt lab and whatever. 144 00:17:12.118 --> 00:17:15.568 Okay. 145 00:17:15.568 --> 00:17:22.318 And by the way, random does not mean completely, uniformly spread out. 146 00:17:22.318 --> 00:17:35.038 I mean, we had that exercise, you know, if something is happening 12 times a year, what are their probabilities that each month gets? 1? Occurrence of the 12 occurrence is actually fairly small. So. 147 00:17:35.038 --> 00:17:40.409 12 random things are happening. They're not going to happen once per month 1 in each separate month. 148 00:17:40.409 --> 00:17:43.709 Probably, they're going to bunch up a little, just randomly. 149 00:17:43.709 --> 00:17:47.159 So, okay simulation. 150 00:17:47.159 --> 00:17:51.838 I'd like to do an example or 2 just to nail down some of this. 151 00:17:51.838 --> 00:17:59.128 Okay. 152 00:17:59.128 --> 00:18:06.419 Starting to get a nice system here by the way. 153 00:18:07.858 --> 00:18:14.429 So, it should be 3 to review and then 1, here my. 154 00:18:14.429 --> 00:18:23.723 I don't know that there's a point in uploading my notes my handwritten notes, because on their own, and not so useful you really have to look at them in conjunction with the recording of the lecture. 155 00:18:24.324 --> 00:18:35.064 So, and since the lecture recording is recording the notes here in this window, I'm thinking there's nothing added by uploading the note file itself. But. 156 00:18:35.398 --> 00:18:39.239 I'm recording. 157 00:18:39.239 --> 00:18:47.878 So, I just saw another claim we'll have a full hour and 20 minutes. That's for the just looking at andrew's question. 158 00:18:49.919 --> 00:18:53.848 Yes, but I want to make a charter. Yeah. 159 00:18:54.868 --> 00:19:01.888 Is there an argument? Not for that accommodations are separate thing so. 160 00:19:01.888 --> 00:19:05.608 Ok, so you draw a deck of cards, you put it back. 161 00:19:05.608 --> 00:19:09.868 And reshuffled says, he might say 2 dependent thing. 162 00:19:09.868 --> 00:19:19.709 So much say what 2 events occur and say would be this is a very simple thing. So. 163 00:19:23.338 --> 00:19:29.038 Card is heart and P equals 2nd card. 164 00:19:29.038 --> 00:19:39.058 Okay, so probability of a there's a 13rd, a quarter of the deck is hard. So. 165 00:19:42.479 --> 00:19:45.659 Okay, and then. 166 00:19:47.759 --> 00:19:51.898 So the probability and let's say see is exactly. 167 00:19:53.548 --> 00:19:57.778 1 heart. 168 00:19:57.778 --> 00:20:04.618 So C is going to be like, you know, a, and not be, or, you know, not a, and B. 169 00:20:04.618 --> 00:20:11.219 And so probability you see, these are just joined so. 170 00:20:11.219 --> 00:20:22.469 Um, so probability of a is 1 corner probability of not be 3 quarters, and I can pull it because they're just dropping the occurrences here. And then not a. 171 00:20:24.328 --> 00:20:27.838 And that's going to be. 172 00:20:29.068 --> 00:20:33.509 616th 3, that's on there. 173 00:20:34.739 --> 00:20:40.618 And this works, because it's drawing with replacement, I put it back and re, shuffle. 174 00:20:40.618 --> 00:20:47.249 Now, number 2 here, so I draw a card. I look at it. I put it back. 175 00:20:47.249 --> 00:20:53.939 The 2nd card is the 1st time I see hearts. 176 00:20:53.939 --> 00:20:57.209 So this means. 177 00:20:57.209 --> 00:21:03.148 If you look at this, so the 1st draw is not a hard. 178 00:21:05.969 --> 00:21:11.128 And the probability there is 3 quarters and then the 2nd. 179 00:21:11.128 --> 00:21:16.618 Draw is art and it's 1 quarter. 180 00:21:16.618 --> 00:21:20.608 So, the probability here is going to be. 181 00:21:20.608 --> 00:21:29.398 3 quarters, 1 quarter equals 16 so that the 2nd card is the 1st time. I see hearts. So. 182 00:21:30.419 --> 00:21:35.939 3, let's see page 88 so longer question here. 62. 183 00:21:45.209 --> 00:21:51.449 Okay. 184 00:21:52.499 --> 00:21:55.949 Oops. 185 00:21:57.269 --> 00:22:03.088 So trying to make a touch bigger to see if I can help people out here. 186 00:22:05.038 --> 00:22:10.019 That's still being. 187 00:22:11.818 --> 00:22:17.338 Okay, good. 188 00:22:17.338 --> 00:22:21.028 So, we're tossing the die twice and. 189 00:22:22.648 --> 00:22:26.818 Number of gods facing up. 190 00:22:28.348 --> 00:22:41.638 Ok, starting page here. 191 00:22:41.638 --> 00:22:47.638 No, it's good enough here. Okay. How would we do that? 192 00:22:54.778 --> 00:22:58.558 Well, the easy case is that. 193 00:22:58.558 --> 00:23:12.959 To probably be, that's 1, 6 number dots in the 1st toss is 6 number of dots and the 1st toss is not less than the number of dots in the 2nd toss. 194 00:23:12.959 --> 00:23:15.959 What I would do here is just. 195 00:23:17.038 --> 00:23:22.439 Graph occurrences, so for a, we would have something like. 196 00:23:25.199 --> 00:23:29.969 Plot things, um, so we've got 1 2. 197 00:23:29.969 --> 00:23:35.638 I'm doing graph paper here, but you can't really see it. So, let me just add Shelly. 198 00:23:37.888 --> 00:23:43.078 We just label it here actually. 199 00:23:44.519 --> 00:23:48.058 Um. 200 00:23:48.058 --> 00:23:51.538 You could just draw it. 201 00:23:59.398 --> 00:24:11.969 Okay, so a number of Docs in the 1st toss, so this would be 1st toss and this will be 2nd and I just write them out. 202 00:24:11.969 --> 00:24:17.429 1st, not less than a number of dots and the 2nd, so what are the, a thing here? 203 00:24:17.429 --> 00:24:21.088 Um, well. 204 00:24:24.058 --> 00:24:32.009 Not so if the 2nd toss was 1, then the 1st toss. 205 00:24:32.009 --> 00:24:36.568 It'd be anything 6 of them if the 2nd task. 206 00:24:36.568 --> 00:24:40.769 I need to draw a line. 207 00:24:40.769 --> 00:24:46.078 That was this 1. 208 00:24:48.509 --> 00:24:52.949 Okay, right so if the 2nd toss. 209 00:24:52.949 --> 00:24:56.219 Was a 1, then the 1st task. 210 00:24:56.219 --> 00:25:04.199 It'd be anything from 1 up as the 2nd toss was to the 1st task. 211 00:25:05.398 --> 00:25:12.118 The 2nd toss was 3. the 2nd task was 4 is 1st, 1 before 5 or 6. 212 00:25:12.118 --> 00:25:19.499 Okay, what is this the summer from 1 to 6? Then that would be. 213 00:25:19.499 --> 00:25:25.618 Would be 21 so so the probability of a here. 214 00:25:26.729 --> 00:25:32.519 2136 which we could reduce down to. 215 00:25:35.489 --> 00:25:42.419 Okay, now the probability of a given be, how would we do that? 216 00:25:44.519 --> 00:25:50.699 Well, we could do that as probably if they and be. 217 00:25:50.699 --> 00:25:54.179 Divide by probably be or something. 218 00:25:54.179 --> 00:26:00.868 Well, a, and B, so this is this is a here a, and B. 219 00:26:00.868 --> 00:26:04.048 So, I'm going to red here. 220 00:26:04.048 --> 00:26:10.048 B, is 1st toss is 6. that's the right hand column here. 221 00:26:10.048 --> 00:26:14.548 So, this here. 222 00:26:14.548 --> 00:26:21.239 Is that is, um, event B actually. Okay. Um. 223 00:26:22.378 --> 00:26:25.439 And a, and to be. 224 00:26:25.439 --> 00:26:29.009 Well, bees a subset of a actually. 225 00:26:29.009 --> 00:26:32.519 So, a, and B. 226 00:26:32.519 --> 00:26:38.249 Is 1, 6 divided by probability of B. 227 00:26:41.159 --> 00:26:46.798 Which is 1 6 actually, um. 228 00:26:49.919 --> 00:26:53.278 I don't know sorry here I got it backwards. 229 00:26:55.523 --> 00:26:56.273 2nd 230 00:27:10.074 --> 00:27:10.703 K. 231 00:27:12.148 --> 00:27:18.328 We're in my back in here. 232 00:27:19.499 --> 00:27:22.528 Okay, sorry B, being 6. 233 00:27:22.528 --> 00:27:26.969 No, it was right, right. 234 00:27:29.338 --> 00:27:35.398 I was right what am I doing? Mind is going. Okay this is in fact. 235 00:27:35.398 --> 00:27:38.999 Be here to 6. 236 00:27:38.999 --> 00:27:47.159 I was right. Okay. And a, that the event in the 1st toss. 237 00:27:50.638 --> 00:27:56.308 Yeah, so he's a subset of B. okay. Just a. 238 00:27:56.308 --> 00:27:59.788 So, a, and B. 239 00:27:59.788 --> 00:28:05.729 That's 1 6 I was right I was getting myself confused. 240 00:28:07.048 --> 00:28:12.449 So, if the number of, um. 241 00:28:17.669 --> 00:28:26.939 If the number of dots in the 1st office is 6 then is certainly not less. The number of dots in the 2nd, talk to a given B is 1. 242 00:28:26.939 --> 00:28:30.179 Probability of be given a. 243 00:28:31.679 --> 00:28:34.919 Well, I'll put it on the same page here. 244 00:28:37.439 --> 00:28:43.648 So probability say a, and B, because I have a probability of a. 245 00:28:43.648 --> 00:28:47.338 Be given a. 246 00:28:47.338 --> 00:28:53.519 That's going to be 1, 6 divided by the probability of a I said was 712. 247 00:28:54.808 --> 00:29:00.628 On 6 times, 12, 7 sequels, 2 sevens. 248 00:29:00.628 --> 00:29:03.868 So so he can do that sort of thing. 249 00:29:03.868 --> 00:29:08.489 Okay, um. 250 00:29:08.489 --> 00:29:13.858 Let me look at 276 and maybe 2 1. 251 00:29:16.888 --> 00:29:24.509 Oh, 1st, I'll leave it just to set up for. Let me go back to that and and. 252 00:29:24.509 --> 00:29:28.348 Okay, this is a question 62. let me let it sit for a minute. 253 00:29:28.348 --> 00:29:32.699 To see, if anyone is totally confused, but. 254 00:29:36.239 --> 00:29:45.659 Does the sum from 1 to? I. okay, Jake was for the number of partitions. 255 00:29:50.669 --> 00:29:55.858 Give me a context for that question Andrew. 256 00:29:57.778 --> 00:30:02.939 Um. 257 00:30:02.939 --> 00:30:07.648 It was to be. 258 00:30:11.608 --> 00:30:15.328 To be on. 259 00:30:16.528 --> 00:30:20.249 In the book on my notes and it doesn't. Okay. 260 00:30:20.249 --> 00:30:24.209 And 1, 2, 3. 261 00:30:24.209 --> 00:30:28.019 There is a summation thing with. 262 00:30:28.019 --> 00:30:34.528 combinate tourists, we have something here. 263 00:30:34.528 --> 00:30:39.209 There is something like come on. 264 00:30:41.249 --> 00:30:50.699 It's not synchronized. 265 00:30:50.699 --> 00:30:53.699 We get that. 266 00:30:53.699 --> 00:31:00.868 Is 1 summation thing it's not exactly partitions. 267 00:31:02.729 --> 00:31:07.229 Above section 2.4. 268 00:31:07.229 --> 00:31:11.759 Oh, in previous sorry. Okay. I wasn't looking back here. 269 00:31:11.759 --> 00:31:20.068 Let me. 270 00:31:44.219 --> 00:31:48.929 Yeah, okay for a here. 271 00:31:48.929 --> 00:31:52.169 So this is. 272 00:31:53.669 --> 00:31:57.269 Yeah, um. 273 00:32:00.269 --> 00:32:05.999 Be 1 here. 2.61 yeah. Um. 274 00:32:05.999 --> 00:32:10.078 So, what's happening here is that. 275 00:32:15.834 --> 00:32:30.624 So there is. 276 00:32:33.328 --> 00:32:39.719 Okay, how we would prove that. 277 00:32:42.628 --> 00:32:47.429 Approved and talked about it, I'm going to assume that that part is true, but. 278 00:33:21.088 --> 00:33:27.628 And then the next is hoops. 279 00:33:29.278 --> 00:33:33.058 And cetera, okay. 280 00:33:43.679 --> 00:33:50.189 Um. 281 00:33:51.838 --> 00:34:04.769 Um, then it's going to be. 282 00:34:04.769 --> 00:34:10.559 And if you work out the formula. 283 00:34:10.559 --> 00:34:14.938 Then it's. 284 00:34:16.798 --> 00:34:23.518 Silence. 285 00:34:26.159 --> 00:34:35.728 Silence. 286 00:34:41.998 --> 00:34:49.559 And then. 287 00:34:49.559 --> 00:34:53.278 Silence. 288 00:34:55.739 --> 00:34:59.998 Where, and equals the sum of all the K. 289 00:35:02.099 --> 00:35:05.818 Is that totally confusing or. 290 00:35:08.938 --> 00:35:13.438 Is the sum of all the Kay? Yes, it is does is equal to yes. 291 00:35:13.438 --> 00:35:17.039 Okay, and. 292 00:35:22.528 --> 00:35:25.619 And that's showing that number of partition. Yeah. 293 00:35:26.668 --> 00:35:34.469 Yeah, okay. Yeah. Okay. So. 294 00:35:34.469 --> 00:35:39.449 Going to do another 1. let me see 276. 295 00:35:43.469 --> 00:35:46.528 Okay, here. 296 00:35:46.528 --> 00:35:50.398 So. 297 00:35:52.199 --> 00:35:59.518 You know, relevant problem. So we've got a 100 items is an unknown number are defective. 298 00:35:59.518 --> 00:36:12.568 And we assume that the defects are mixed in randomly that we're pulling them randomly out of this. What I mean is, we are not, for example, picking the 1st, 2 that were produced. 299 00:36:12.568 --> 00:36:18.268 Because maybe the machine was unusually gutter and usually bad at the start. So we're picking randomly. 300 00:36:18.268 --> 00:36:21.389 I never generator or whatever and. 301 00:36:23.039 --> 00:36:28.829 And doing the tests might be expensive and doing the tests might destroy the object. 302 00:36:28.829 --> 00:36:35.759 Okay, so we don't want to know. So if we test all 100 items fine, but now we've got no items left to sell. 303 00:36:35.759 --> 00:36:40.018 And but, you know, we test 2 items. 304 00:36:40.018 --> 00:36:43.889 And then, you know, if they're. 305 00:36:43.889 --> 00:36:46.918 If 1 of them is bad, then. 306 00:36:48.898 --> 00:36:55.318 We reject the whole lot, or we accept it. But the problem is that. 307 00:36:55.318 --> 00:37:03.179 We may be accepting a lot that has some bad items, and we send them to the customer and the customer gets angry. 308 00:37:03.179 --> 00:37:08.518 So, what do we do here? How would we handle this? 309 00:37:11.909 --> 00:37:15.239 Build it a lot with K items. 310 00:37:15.239 --> 00:37:20.668 Effective items. Well. 311 00:37:20.668 --> 00:37:25.018 And problem. 312 00:37:26.159 --> 00:37:31.349 76, hey, well, so the probability. 313 00:37:31.349 --> 00:37:42.358 Of a bad item over 100 I mean, that is stated okay that stated that the law has paid effective items. Now, given that. 314 00:37:42.358 --> 00:37:49.559 Call that just P, now, given that we're going to test 2 items and. 315 00:37:49.559 --> 00:37:58.469 What's the probability that 1 of them is defective? Well, I could just it's easy to say. What's the probability that they're both good? 316 00:37:58.469 --> 00:38:01.889 Um. 317 00:38:01.889 --> 00:38:10.228 Both good is going to be 1 minus P that the 1st 1 is good and then squared. 318 00:38:10.228 --> 00:38:16.619 So. 319 00:38:18.780 --> 00:38:26.579 So this is that if we have K, defective items, this is a probability that we will. 320 00:38:26.579 --> 00:38:33.210 Except the law, it is Kate effective items, but purely by chance when we tested 2. 321 00:38:33.210 --> 00:38:36.989 They were both good. So E. G. 322 00:38:38.730 --> 00:38:47.219 I'm not suppose there are, I suppose there's 10 of the 100 are bad. 323 00:38:47.219 --> 00:38:52.920 And here, our probability is going to be probability. We, except a lot. 324 00:38:52.920 --> 00:38:57.480 Equals point 9 squared. It was. 325 00:38:57.480 --> 00:39:02.909 Point 8, 1, by just testing 2 of them. 326 00:39:06.090 --> 00:39:11.460 Now, be good. Interesting. 327 00:39:12.480 --> 00:39:20.550 So, we have a really bad production line here have. 328 00:39:20.550 --> 00:39:23.639 Of the items that it makes are bad. 329 00:39:25.500 --> 00:39:32.610 And some be here, so probably is a bad item is 1 half. 330 00:39:32.610 --> 00:39:36.809 Okay says all right. 331 00:39:36.809 --> 00:39:43.079 It's bad it's horrible, but now, if we test this. 332 00:39:44.519 --> 00:39:50.489 Well, the question is, we have to design a test so that if we test perhaps. 333 00:39:50.489 --> 00:39:54.030 5 items or something, and if we find a defect. 334 00:39:54.030 --> 00:40:05.340 That in this case, 99% chance that we will find at least find at least 1 defect. So, inside your different words, on what we're going to do is, we. 335 00:40:07.289 --> 00:40:15.510 When a test K items and, um, let's say, and, um. 336 00:40:15.510 --> 00:40:19.500 He's a different letter here. I don't want to confuse you. 337 00:40:22.199 --> 00:40:26.010 Test and items and. 338 00:40:32.130 --> 00:40:37.949 What is in? So that. 339 00:40:41.489 --> 00:40:45.809 All and items, um. 340 00:40:47.010 --> 00:40:53.519 Test good. 341 00:40:53.519 --> 00:40:59.699 Probability less 1. okay so then they're going to find 1 or more items. 342 00:40:59.699 --> 00:41:07.110 Bad 99% chance. So that's another way of saying is that we test and items and then the probability. 343 00:41:07.110 --> 00:41:10.920 Of being all good is not more than 1%. 344 00:41:10.920 --> 00:41:16.349 Well, this is 50 50 that an item is bad so that the. 345 00:41:18.630 --> 00:41:27.690 Basically, the probability that all and items are good is on half to the end and we want to know. 346 00:41:30.059 --> 00:41:37.409 And so basically, 1 and equal 7, if we test 7 items, it's 1 over 2 to the 7th that they were all good. 347 00:41:37.409 --> 00:41:47.010 So so, if we want only a 1% chance of not noticing this bad lot, we've got to test 7 items out of the 100. 348 00:41:48.599 --> 00:41:59.190 Makes sense that you think about that for a 2nd, here is the 276. this is a fun question. So. 349 00:42:02.130 --> 00:42:07.980 Professor, yes. Or a number seems high. Am I reading that wrong? 350 00:42:07.980 --> 00:42:11.789 Or not. 351 00:42:11.789 --> 00:42:15.719 So, what does that 81% mean? 352 00:42:15.719 --> 00:42:19.170 Oh, we're up here. 353 00:42:20.280 --> 00:42:28.050 Here so. 354 00:42:31.679 --> 00:42:42.150 Okay, I added an, a, I added something is that I, to a here I said what, if 10% of the items are bad. Okay. 355 00:42:42.150 --> 00:42:51.090 Um, so Kay is 10 out of a 100 K was 10 here. So I said if K was using a sift as if K is 10. 356 00:42:51.090 --> 00:42:57.989 And we use this test. What's the probability that we erroneously accept? The lot. 357 00:42:57.989 --> 00:43:02.489 But so we checked the 1st item. 358 00:43:02.489 --> 00:43:07.769 90% chance that it passes, because only 10% of the items are bad. 359 00:43:07.769 --> 00:43:11.429 The 2nd, we test the 2nd item. 360 00:43:11.429 --> 00:43:18.269 90% chance that passes so 81% chance that both items are good. 361 00:43:18.269 --> 00:43:21.539 Even though, 10% of all, the items are bad. 362 00:43:23.579 --> 00:43:27.539 I think the 81 is correct. Good question actually about that. 363 00:43:27.539 --> 00:43:31.050 Yes, for both them being good. Um. 364 00:43:31.050 --> 00:43:35.130 It's key over a 100 right? But when you choose the 1st, 1. 365 00:43:35.130 --> 00:43:43.289 Wouldn't it be out of 99 for this? 2nd? Oh, I see what you're getting at selection without replacement. 366 00:43:43.289 --> 00:43:48.269 Um, thank you. You're right on that. 1. 367 00:43:49.590 --> 00:43:53.699 Send me email I'll give you a point. Yeah. 368 00:43:56.519 --> 00:44:04.260 You're right, it's not an incredible difference, but what I did is wrong in that sense. Yeah. Let me write that down for the rest of, you. 369 00:44:04.260 --> 00:44:07.710 We were talking about part a here. 370 00:44:10.260 --> 00:44:17.190 Eva. 371 00:44:41.610 --> 00:44:47.460 Right Thank you. Ok, so. 372 00:44:50.789 --> 00:44:55.170 So, effective items, it's, um. 373 00:44:56.880 --> 00:45:01.139 K over 100 that the 1st test item fails. 374 00:45:01.139 --> 00:45:04.619 And K minus 1 over 99. 375 00:45:07.500 --> 00:45:13.500 That I'm doing this passes. Sorry I'm just a 2nd here. 376 00:45:21.300 --> 00:45:28.260 So the 1st, 1 passes was probability 100 minus K over 100. 377 00:45:28.260 --> 00:45:32.940 2nd 1 passes 99. 378 00:45:34.409 --> 00:45:37.829 And it's going to be conditional on the 1st 1, let's say. 379 00:45:37.829 --> 00:45:41.880 So we got 99 and there are still Kay bad ones. 380 00:45:44.820 --> 00:45:54.630 Okay, it's a conditional thing actually, because we don't do the 2nd test. 381 00:45:54.630 --> 00:45:58.260 Is condition on it at past the 1st test so. 382 00:45:59.340 --> 00:46:09.090 1st test past and then given it the 1st test pass the 2nd test pass. That's that's what I would get for the answer there. Now Thank you. 383 00:46:13.110 --> 00:46:18.000 That makes sense. So. 384 00:46:18.000 --> 00:46:25.949 Okay. 385 00:46:25.949 --> 00:46:32.789 And the 2nd, 1, does that change the answer? 386 00:46:32.789 --> 00:46:37.829 Because that's not an enormous difference, but it is a difference. That's correct. So. 387 00:46:37.829 --> 00:46:43.530 Silence. 388 00:46:43.530 --> 00:46:57.659 Yeah, um, and actually to do the 2nd, 1 correctly, what we should be doing is. 389 00:46:57.659 --> 00:47:02.429 So. 390 00:47:07.110 --> 00:47:12.090 You know, the 1st test path says test passes. 391 00:47:13.110 --> 00:47:21.179 Is 1 half for the 2nd test. 392 00:47:22.289 --> 00:47:25.619 Well. 393 00:47:27.059 --> 00:47:35.010 The 50 out of 100, it's going to be the same as 51 hundreds of probability. 394 00:47:35.010 --> 00:47:39.300 We're not saying that there is precisely a 100. we've tested 50. 395 00:47:41.159 --> 00:47:50.429 So, the answer I gave for B, would still be the same. So, for for be here and some. 396 00:47:50.429 --> 00:47:56.789 Because I think the reasonable reading is, is the 50. 397 00:47:56.789 --> 00:48:01.050 Of 100 probability. 398 00:48:02.460 --> 00:48:06.480 You know. 399 00:48:08.610 --> 00:48:14.070 It's not specifically. 400 00:48:14.070 --> 00:48:17.400 100 items I think. 401 00:48:17.400 --> 00:48:21.809 Yeah. 402 00:48:21.809 --> 00:48:28.110 Okay, oh, yeah. Just still the same. 403 00:48:28.110 --> 00:48:34.110 1, 2006. 404 00:48:35.670 --> 00:48:41.639 Silence. 405 00:48:49.019 --> 00:48:59.280 Okay. 406 00:48:59.280 --> 00:49:03.809 Well, I'll just give you a summary of it here. I'd say someone. 407 00:49:03.809 --> 00:49:09.510 So, okay, so. 408 00:49:15.599 --> 00:49:21.900 So, what we're doing is a, is sending 2 messages to be. 409 00:49:21.900 --> 00:49:32.190 Okay, it's me. 410 00:49:35.309 --> 00:49:42.750 Okay, and and you hope 1 is and you hope. 411 00:49:45.030 --> 00:49:48.630 At 1 is good. Okay. 412 00:49:48.630 --> 00:50:00.630 So, and the probability of an error is and the messages I checked some to be can tell if they arrived. 413 00:50:00.630 --> 00:50:05.699 Is good. So. 414 00:50:05.699 --> 00:50:09.780 So, the probability that. 415 00:50:13.800 --> 00:50:24.179 So, the probability that both are bad. 416 00:50:25.710 --> 00:50:29.670 Q, one's a probability of an error so that. 417 00:50:31.619 --> 00:50:35.400 Okay, so this is a geometric. 418 00:50:35.400 --> 00:50:39.300 Just just a geometric probability thing. Um. 419 00:50:41.070 --> 00:50:49.949 It is to be a Shannon and. 420 00:50:49.949 --> 00:50:53.820 Probability of success. 421 00:50:56.190 --> 00:50:59.219 On any on any trial. 422 00:51:00.960 --> 00:51:11.489 Is, and you keep going to you get a success. Okay. 423 00:51:11.489 --> 00:51:18.150 And then you can work it out from here. So, and. 424 00:51:19.559 --> 00:51:25.230 Right. 425 00:51:25.230 --> 00:51:30.329 You want to go to and also I'm going to ask you to. 426 00:51:30.329 --> 00:51:34.079 I'm sorry, I didn't hear you. 427 00:51:34.079 --> 00:51:40.650 It would be 1 minus Q1, Q2 and then also 1 minus Q2 Q1. 428 00:51:40.650 --> 00:51:44.099 Right. Why. 429 00:51:48.059 --> 00:51:51.269 Let's say, for example, Q1. 430 00:51:51.269 --> 00:51:55.679 Has a like a 99% rate and then Q2. 431 00:51:55.679 --> 00:51:58.829 Has a point 0, 0, 1 acceptance rate. 432 00:51:58.829 --> 00:52:08.699 The probability of success on any trial is a higher, depending if it's Q1, Q2 the 1 transmitting Q1, Q2 or unnecessarily the same, right? 433 00:52:08.699 --> 00:52:13.679 Oh, no, that's what makes it interesting. They could be different, but they're. 434 00:52:13.679 --> 00:52:18.960 Transmitting the 2 messages simultaneously a, and B, simultaneously. 435 00:52:18.960 --> 00:52:26.099 It's not, it's not a sequence here where we transmit over the 2nd link. Only its 1st link fails. 436 00:52:26.099 --> 00:52:29.909 So that. 437 00:52:29.909 --> 00:52:34.139 So, I'm working here the probability that both links failed. 438 00:52:35.519 --> 00:52:39.869 So, Q1 is the. 439 00:52:39.869 --> 00:52:44.309 Probability of an error so the probability that they were both wrong. 440 00:52:44.309 --> 00:52:48.449 Is Q1. 441 00:52:48.449 --> 00:52:53.730 Q2 that probably is for 1st, 1 is wrong times of probably the 2nd, 1 is wrong. 442 00:52:53.730 --> 00:52:58.469 And probably not both wrong is in 1 minus Q1 Q2. 443 00:52:59.730 --> 00:53:02.969 Does that make sense? 444 00:53:02.969 --> 00:53:06.989 I'm going to reframe what are the probability of success on any trial? 445 00:53:06.989 --> 00:53:11.250 Would be 1 minus Q1 Q2 I think. 446 00:53:13.349 --> 00:53:17.550 Because I get it wrong. Oh, nevermind nevermind I said, I don't know. Yeah. 447 00:53:17.550 --> 00:53:20.880 I misread your handwriting was there. 448 00:53:20.880 --> 00:53:26.039 Sorry about that you can read my handwriting. Okay. Okay. 449 00:53:27.360 --> 00:53:36.119 So, and then so this is geometric for them. We can find that probably the more than K are required. So. 450 00:53:39.809 --> 00:53:44.010 And and then if. 451 00:53:44.010 --> 00:53:48.510 The, at least 1 was good. 452 00:53:50.219 --> 00:53:58.650 Then, I mean, I could continue on with this, but I'll drop this, but then the next 1 probability that. 453 00:53:58.650 --> 00:54:05.130 Link to is good given that at least 1 was good. 454 00:54:06.420 --> 00:54:14.010 I could work that out, so I'll try to write more slowly than probability. 455 00:54:16.710 --> 00:54:28.530 That. 456 00:54:30.150 --> 00:54:44.760 Okay, so the probability that link 2 was good. 457 00:54:48.989 --> 00:54:52.949 That would make you 1 minus Q2. 458 00:54:54.630 --> 00:55:05.489 Probability that at least 1 was good. Is. 459 00:55:09.539 --> 00:55:17.340 And so the probability that. 460 00:55:32.159 --> 00:55:35.969 Right now actually, is. 461 00:55:35.969 --> 00:55:40.949 The probability that too is good. We can't see we are ready. 462 00:55:42.030 --> 00:55:47.880 Oh, I'm sorry, but there okay. 463 00:55:47.880 --> 00:55:55.500 The thing that sorry about that does the synchronize automatically always. So the probability that. 464 00:55:56.699 --> 00:56:03.389 That link too was good, given that at least 1 of the 2 lengths was good. I mean, maybe write this out in my detail. 465 00:56:09.869 --> 00:56:19.260 Of the links was good is probably the 2 good and at least 1 good. 466 00:56:22.769 --> 00:56:29.519 Or, at least 1 good over the probability that at least 1 good. 467 00:56:32.639 --> 00:56:35.820 Now too good. 468 00:56:37.679 --> 00:56:46.530 Well, if too good, that at least 1 is good. So this at least 1 good is a subset of that. So it's too good. 469 00:56:47.730 --> 00:56:53.039 Or, at least 1. 470 00:56:53.039 --> 00:56:59.070 And that's the 2nd, here. 471 00:57:00.630 --> 00:57:09.840 And that's okay and that's going to be the 2 is good. Is. 472 00:57:14.309 --> 00:57:20.670 If I got it, right so I'll let that up for a. 473 00:57:20.670 --> 00:57:30.239 Minnesota, let people think about that. 474 00:57:40.739 --> 00:57:47.309 Can you explain the to good and at least 1 good going probability that too is good. 475 00:57:47.309 --> 00:57:51.090 The subset thing. Okay, sure. Um. 476 00:57:58.920 --> 00:58:03.420 Yeah, good. 477 00:58:05.369 --> 00:58:13.920 At least 1 link is good. 478 00:58:18.420 --> 00:58:22.679 It's fine to, um. 479 00:58:22.679 --> 00:58:26.099 Therefore. 480 00:58:34.739 --> 00:58:38.789 It's a subset so. 481 00:58:45.539 --> 00:58:50.670 So the set to good. 482 00:58:53.400 --> 00:58:57.570 And which is the intersection thing, maybe I'll draw it like that. 483 00:58:59.429 --> 00:59:04.710 Uh, huh. And at least 1. 484 00:59:05.880 --> 00:59:10.110 Equals to set. 485 00:59:10.110 --> 00:59:14.460 Good by 2 I mean. 486 00:59:14.460 --> 00:59:18.719 By 2, I mean, linked too good. So. 487 00:59:18.719 --> 00:59:29.940 Does that make sense if link 2 is good then at least 1 link is good too. So the set to a good and at least 1 link to that's the set too. Good. 488 00:59:29.940 --> 00:59:33.360 So, the probabilities are the same because the sets at the same. 489 00:59:37.949 --> 00:59:41.429 Let me give you an example. 490 00:59:50.760 --> 00:59:57.269 Come on good. Some set. 491 01:00:01.530 --> 01:00:08.489 So di, S1 a close. 492 01:00:08.489 --> 01:00:17.489 Eva number set 2. 493 01:00:17.489 --> 01:00:20.699 Equals. 494 01:00:24.059 --> 01:00:29.880 Kind of know. 495 01:00:29.880 --> 01:00:33.719 Even Square, or something I'm making this up as I'm going along. 496 01:00:33.719 --> 01:00:38.519 Set 4 so, um. 497 01:00:39.690 --> 01:00:42.809 So, that too is a subset of sat 1. 498 01:00:42.809 --> 01:00:49.739 So, the, um, so, um. 499 01:00:49.739 --> 01:00:54.989 To 2 intersect set 1. 500 01:00:54.989 --> 01:01:02.820 It was sent to. Okay, so the probability of set 2 equals the set. 501 01:01:02.820 --> 01:01:06.000 2 intersects at 1. 502 01:01:06.000 --> 01:01:11.550 Um, so. 503 01:01:13.530 --> 01:01:17.340 So, the probability is set to given that 1. 504 01:01:19.079 --> 01:01:29.369 Set to intersect, sent 1 divided by probability to set 1 equals the probability of set to progress at 1. 505 01:01:29.369 --> 01:01:38.610 Okay. 506 01:01:40.050 --> 01:01:41.244 Does that makes sense 507 01:01:57.114 --> 01:01:58.704 in the Venn diagram? 508 01:02:00.900 --> 01:02:05.699 Well, I hear the Venn diagram would be, um. 509 01:02:12.809 --> 01:02:18.539 Sent 1, and then sent 2 sorry backwards. 510 01:02:25.409 --> 01:02:28.710 Set to and set 1. 511 01:02:31.440 --> 01:02:36.570 And I could label everything here. Let me just label everything. This is so set. 512 01:02:36.570 --> 01:02:46.710 2 is the 4 that 1 is 2, 4 and 6 that's also 2 and 6 and outside are 1, 3 and 5. 513 01:02:46.710 --> 01:02:50.369 And this would be the whole big universe hair. 514 01:02:52.110 --> 01:02:59.280 The whole universe. Okay. 515 01:03:01.920 --> 01:03:10.079 Makes sense. 516 01:03:10.079 --> 01:03:14.340 Silence. 517 01:03:15.719 --> 01:03:25.860 Okay, it's a little late to start chapter 3. I just want to give you a. 518 01:03:25.860 --> 01:03:30.539 Teaser for it right now and then. 519 01:03:34.170 --> 01:03:37.710 And let me see, what did I type up for it here? 520 01:03:37.710 --> 01:03:40.860 Oh, 1 more thing. 521 01:03:45.030 --> 01:03:54.300 We can okay, I don't I would not trust Wikipedia for politics or current offense, but I do find them quite. 522 01:03:54.300 --> 01:03:58.559 Useful for technical topics. 523 01:03:59.639 --> 01:04:03.389 And read about them for random variables, for example. 524 01:04:03.389 --> 01:04:14.550 In fact, there almost more detailed, so they've got examples with coins and dice discreet, continuous, mixed type. 525 01:04:14.550 --> 01:04:24.119 And so this is all deeper than you want almost but this is a useful place, perhaps, you know, to see some fun stuff. 526 01:04:24.119 --> 01:04:27.840 Outcomes and that sort of thing. So. 527 01:04:27.840 --> 01:04:32.519 Oops, that 1 didn't work anymore and link. Oh, okay. 528 01:04:32.519 --> 01:04:46.590 Fun with something and that's enrichment and this testing error thing is enrichment for the moment. Also. I mentioned it because you see it with current events yeah. 529 01:04:46.590 --> 01:04:53.670 Talking about tests for children and etc sensitivity tests and selectivity tests and so on. 530 01:04:53.670 --> 01:05:03.300 Okay, so what will happen in chapter 3 just to give you a quick teaser. 531 01:05:03.300 --> 01:05:08.039 Is we're looking at more discrete random variables so we. 532 01:05:08.039 --> 01:05:14.070 Seeing you and this is also this start here stuff we've seen in uniform. 533 01:05:14.070 --> 01:05:17.699 We're going to see different types of uniform, random variables. 534 01:05:17.699 --> 01:05:26.280 Tossing fair tossing a dye will doesn't have equal. Probably it's fair. We're newly tossing a coin. 535 01:05:26.280 --> 01:05:32.159 Binomial tossing a coin and times and looking at the number of heads. We don't care. 536 01:05:32.159 --> 01:05:37.139 What order the heads came and, and a multi nomial. 537 01:05:37.139 --> 01:05:40.710 We're tossing the dye and times so we're looking at the number of. 538 01:05:40.710 --> 01:05:44.789 Times of 1 came up the number of times the 2 came up up to 6, and we don't care. 539 01:05:44.789 --> 01:05:49.469 What order they came and just the number that came up and geometric. 540 01:05:49.469 --> 01:05:53.190 And so on now. 541 01:05:53.190 --> 01:06:00.840 What happens in chapter 3 is we take these discreet random. We see other examples of that. 542 01:06:00.840 --> 01:06:13.050 And we said, well, another example, something called a croissant, for example, we are looking at, say, cosmic race hitting your experiment and. 543 01:06:13.050 --> 01:06:19.619 That's the run. It's a discrete number cosmic re, sitting your experiment or number you got to grab a Radium. 544 01:06:19.619 --> 01:06:22.920 How many Adam's decay in the next minute. 545 01:06:22.920 --> 01:06:32.909 It's an endangered from 0T op and is a probability distribution and it's a discreet, random variable, for example, and we're going to see things like. 546 01:06:34.380 --> 01:06:40.650 Calculating statistic, expectations working with combinations of random variables. 547 01:06:40.650 --> 01:06:47.250 You may be you're placing a Beth in a casino and your winnings depend on. 548 01:06:47.250 --> 01:06:59.820 You know what your hand with your card show, and it's some non linear function. So it'll be saying stuff like that. So, things like mean and whatever. 549 01:06:59.820 --> 01:07:03.599 Or it would be, or you weren't going to be winning or losing at this game. 550 01:07:03.599 --> 01:07:11.489 Stuff like that, so it would be things things like functions and stuff like that operations on them. 551 01:07:11.489 --> 01:07:15.360 Things called variants and that's how much it spreads out. 552 01:07:15.360 --> 01:07:18.840 And whatever. So. 553 01:07:18.840 --> 01:07:22.440 If you're tossing a coin. 554 01:07:22.440 --> 01:07:29.369 Um, maybe it's not a fair coin or tossing dice. Maybe you'll get paid off more. 555 01:07:29.369 --> 01:07:38.579 If you get 2, sixes, and if you get 2, 5 to something non linear. So the mean, and then the variance would be would work into how likely you are to win. 556 01:07:38.579 --> 01:07:42.329 From how much you would expect to win perhaps. 557 01:07:42.329 --> 01:07:49.320 He'll be saying stuff like that and okay. 558 01:07:49.320 --> 01:07:54.239 Other things, so, um. 559 01:07:55.349 --> 01:07:58.349 Here's a crazy 1. my example here. 560 01:07:58.349 --> 01:08:03.360 You know, you're saying. 561 01:08:03.360 --> 01:08:09.869 If climbing whatever us you paying 70000 dollars at time and. 562 01:08:09.869 --> 01:08:17.579 Each time has 4th chance of succeeding and then what's the expected cost? Do you get to the top. 563 01:08:17.579 --> 01:08:24.420 Oh, then I could make it weirder. Let's suppose on each time you have a 5% chance of dying. 564 01:08:24.420 --> 01:08:27.840 And I present is pretty good. It used to be a lot higher. 565 01:08:27.840 --> 01:08:30.989 So then we could talk about questions. 566 01:08:30.989 --> 01:08:38.310 And you keep, you know, and you keep trying Mount Everest and you either get to the top, or you die. Whichever happens. 1st. 567 01:08:38.310 --> 01:08:43.229 So, now we can talk about what's the probability that you get to the top before you die? 568 01:08:43.229 --> 01:08:48.210 And if that happens, what's your expected cost? 569 01:08:48.210 --> 01:08:52.380 That sort of thing, so, all sorts of things with these. 570 01:08:52.380 --> 01:08:59.699 Okay, so that's what we'll be seeing next time discrete random variables. 571 01:09:01.409 --> 01:09:06.510 Okay, any final. 572 01:09:06.510 --> 01:09:14.460 Questions and again, for the test will run it on grade scope. It will be simple. 573 01:09:14.460 --> 01:09:18.600 Answers may take some thinking. 574 01:09:18.600 --> 01:09:23.939 And I hope is, you're not time constrained. 575 01:09:23.939 --> 01:09:32.640 That's my hope, but I'm not able to predict these things. Exactly. It's the time it takes around and variable. 576 01:09:32.640 --> 01:09:36.569 And so and. 577 01:09:36.569 --> 01:09:40.890 Case are also are having office hours, of course, and been announced. 578 01:09:40.890 --> 01:09:44.220 An earlier thing in the blog. 579 01:09:45.779 --> 01:09:51.510 By the way, if you want to find something in the blog, you can also like the archive here. 580 01:09:51.510 --> 01:09:55.170 The blog is every thing. 581 01:09:55.170 --> 01:09:59.699 Okay, and it's and you can just search here, so. 582 01:10:02.460 --> 01:10:08.250 Okay, so you want to search for office hours you could search for office. 583 01:10:08.250 --> 01:10:13.649 You see that sort of thing it's useful having everything here at once. 584 01:10:16.289 --> 01:10:26.100 Other questions. Okay. 585 01:10:27.779 --> 01:10:31.979 In that case, I'll see you Thursday that. 586 01:10:35.189 --> 01:10:39.840 Actually, this is worth saving here. 587 01:10:42.300 --> 01:10:45.720 So, professor, yes, for the exam. 588 01:10:45.720 --> 01:10:52.229 So, what time is it going to be open? Is it going to be open throughout the entire day or is it going to be open? 589 01:10:52.229 --> 01:10:59.220 On class time, I got 3 o'clock. My intention is to make it open at class time. Actually. 590 01:11:00.300 --> 01:11:03.810 And it's going to be a 1 hour and 20 minutes long. So, like, the entire. 591 01:11:03.810 --> 01:11:08.520 Yes, and then 1 more thing, and then for the format of the test. 592 01:11:08.520 --> 01:11:13.319 You mentioned that it's short answers so you can by that. Do you mean do we write our answers? 593 01:11:13.319 --> 01:11:18.390 On grade scope itself, or do we have to take pictures of our work and upload it on grades? 594 01:11:18.390 --> 01:11:24.090 My hope is to have it that you can write answers in grade scope because that will be easier for, you. 595 01:11:25.890 --> 01:11:30.090 Either way it works, it depends on the question. Like, if it's like. 596 01:11:30.090 --> 01:11:34.289 Asking for work, then writing would be easier for me personally, but. 597 01:11:34.289 --> 01:11:43.680 But, yeah, it's up to you, I guess. Well, okay, I was thinking that having to photograph things as harder, but floor is open. What do other people feel about this? 598 01:11:47.399 --> 01:11:51.810 Okay, all out photographing and so on I guess you got your phone, you. 599 01:11:51.810 --> 01:11:56.729 Snap and upload so okay. 600 01:11:56.729 --> 01:12:02.909 Well, if we allow photographing, then you can, have you writing a little math or something? 601 01:12:05.159 --> 01:12:09.210 Okay, other questions. 602 01:12:10.859 --> 01:12:18.329 Oh, I can't you won't have access to certain assembles. Right? So. 603 01:12:18.329 --> 01:12:22.319 I wouldn't do anything it force you to use cymbals. So. 604 01:12:25.380 --> 01:12:31.680 You know, it would be nice if we had handwriting recognition for mass symbols, but we're not quite there yet. 605 01:12:32.484 --> 01:12:42.114 So, other points, well, let me review it a question. 606 01:12:42.114 --> 01:12:49.944 I asked at the start so what I'm writing here, my handwritten notes since I'm projecting them and recording them. 607 01:12:50.250 --> 01:12:53.819 With the video for the class. 608 01:12:53.819 --> 01:12:59.670 Do you also want me to upload the handwritten notes separately as a PDF or not? 609 01:13:01.140 --> 01:13:06.600 I'll do it if anyone wants it, but I haven't gotten anyone saying you want it. 610 01:13:08.850 --> 01:13:18.449 So, okay, well, if anyone wants them in the future, I can upload other. Like I said, they make more sense if you're looking at the video. 611 01:13:20.760 --> 01:13:28.409 Okay, well. 612 01:13:39.239 --> 01:13:40.470 Okay.