WEBVTT 1 00:00:20.910 --> 00:00:47.399 Silence. 2 00:00:52.770 --> 00:01:00.990 Silence. 3 00:01:09.750 --> 00:01:20.069 Silence. 4 00:01:20.069 --> 00:01:28.290 Okay, so good afternoon class. This is probability. Class 718th. 5 00:01:28.290 --> 00:01:33.390 This 1021 and my universal question is. 6 00:01:33.390 --> 00:01:36.989 Can you hear me? Yep. 7 00:01:36.989 --> 00:01:47.519 Hey, thank you. Beautiful. And I have the chat window open. So, the theory is that you post questions, I may even see them. 8 00:01:47.519 --> 00:01:53.519 So, I have up, um, what is happening today. 9 00:01:56.489 --> 00:01:59.730 1st. 10 00:01:59.730 --> 00:02:10.590 Because of various questions, I added a link up at the top here to the videos, and you can go in and get to media site videos here. 11 00:02:11.819 --> 00:02:21.180 And now they ought to all be readable if they're not let me know. 12 00:02:21.180 --> 00:02:29.550 Remarks and reasonable to the community and then somehow they snap back to private. Only I don't know why I marked some of them. 13 00:02:29.550 --> 00:02:33.689 I've reset 2 permissions twice on some of them, but any case there's. 14 00:02:33.689 --> 00:02:39.389 Videos okay, what's happening today? Is. 15 00:02:39.389 --> 00:02:46.229 Oh, 1st exam 1. we'll set that in a week and a half on March. 1st and. 16 00:02:46.229 --> 00:02:53.789 To accommodate, I mentioned the details later, but to accommodate the students in China who are 12. 17 00:02:53.789 --> 00:02:57.599 Hours of set will run the exam twice. 18 00:02:57.599 --> 00:03:00.719 Might have different questions slightly so that. 19 00:03:02.039 --> 00:03:07.020 You know, people don't have to be up in the middle of the night, which is not fair. 20 00:03:08.069 --> 00:03:13.110 Now, this is a remote exam and. 21 00:03:13.110 --> 00:03:19.379 We have to have some sense of an honor code here. I cannot. 22 00:03:19.379 --> 00:03:33.449 Stop cheating these technological things, which pretend to stop students teaching a, they're very obnoxious and obtrusive and they don't even work very well. So we have to have some concept here. 23 00:03:33.449 --> 00:03:39.360 About you're not supposed to cheat to engineers just supposed to have some internal rules here. 24 00:03:39.360 --> 00:03:51.449 What we'll also do is, I'll do this fictional University exam that I miss example that I messed up last time will try to do it some more. 25 00:03:51.449 --> 00:03:57.330 Some more things, and then we'll continue on with Bayes theorem and independent events. 26 00:03:57.330 --> 00:04:01.979 And we will finish up chapter 2. 27 00:04:03.210 --> 00:04:07.830 And I have here. 28 00:04:07.830 --> 00:04:11.129 Okay. 29 00:04:11.129 --> 00:04:14.460 The theory is that this other toy is working. 30 00:04:14.460 --> 00:04:17.730 Is working somewhat. 31 00:04:18.839 --> 00:04:32.038 Silence. 32 00:04:36.209 --> 00:04:39.869 Come on and what's going on here. 33 00:04:44.249 --> 00:04:58.019 Silence. 34 00:05:06.238 --> 00:05:20.488 Silence. 35 00:05:20.488 --> 00:05:33.059 Okay, so. 36 00:05:34.288 --> 00:05:41.608 Okay, so we did some things some examples. Last time I'll do some more. 37 00:05:41.608 --> 00:05:45.718 And just to. 38 00:05:47.189 --> 00:05:52.108 Let's do some of the complimentary ones let's say. 39 00:05:53.129 --> 00:06:00.658 Probability of say not being an engineer. The 7th you can just count up. It's going to be. 40 00:06:00.658 --> 00:06:08.819 3 sevens, for example, probably of not being an undergrad. There are. 41 00:06:10.649 --> 00:06:16.408 Is 2 segments for example, probability say. 42 00:06:16.408 --> 00:06:21.478 Of not being an engineer, and also not being an undergrad. 43 00:06:21.478 --> 00:06:25.649 That's that white thing outside. That's 1. 7th. 44 00:06:25.649 --> 00:06:31.019 For example, so now we can do things say. 45 00:06:31.019 --> 00:06:35.848 The probability of not being an engineer and given not being an undergrad. 46 00:06:35.848 --> 00:06:41.278 Probably not being an engineer and given. Not me. 47 00:06:51.413 --> 00:06:52.733 Not being an engineer, 48 00:07:07.764 --> 00:07:09.593 so you're not an undergrad. 49 00:07:10.829 --> 00:07:15.238 Then it's a 50, 50 chance you're not an engineer, for example. 50 00:07:15.238 --> 00:07:20.639 We can start adding up, um. 51 00:07:23.338 --> 00:07:27.059 Also, as I probably not being an engineer given. 52 00:07:27.059 --> 00:07:35.519 You are an undergrad and you're an undergrad. Well, that's going to be 2 sevens there. 53 00:07:35.519 --> 00:07:44.759 You're not in say probability of not being engineer is a, probably of not being an engineer. 54 00:07:44.759 --> 00:07:49.619 And not being an undergrad, plus the probability of not being an engineer. 55 00:07:49.619 --> 00:07:53.968 And an undergrad, which is, um. 56 00:07:56.939 --> 00:08:01.379 For example. 57 00:08:02.639 --> 00:08:08.788 And we can do conditional stuff like that. That's also you could also write this. 58 00:08:12.749 --> 00:08:20.488 Given you're not an undergrad, not being an undergrad plus. 59 00:08:22.619 --> 00:08:26.428 You're and undergrad times the probability of an undergrad. 60 00:08:27.629 --> 00:08:34.349 And we could work all that out and so not being an engineer giving you're not an undergrad and. 61 00:08:36.989 --> 00:08:43.918 Is 1, half times of probability of not being an undergrad. 62 00:08:43.918 --> 00:08:47.428 Which was, um, 2, 7th. 63 00:08:49.678 --> 00:08:57.058 Plus, I probably have not been an engineer given your an undergrad is 2 sevens. 64 00:08:57.058 --> 00:09:00.119 The probability of being an undergrad. 65 00:09:00.119 --> 00:09:03.418 Which is. 66 00:09:09.568 --> 00:09:15.688 Is a 2nd, here probably is not being an engineer giving your an undergrad is. 67 00:09:18.568 --> 00:09:29.519 Sorry, 1 is wrong. Probably if not being an engineer if you're an undergrad is going to be. 68 00:09:29.519 --> 00:09:35.219 To fess at, Charlie. 69 00:09:35.219 --> 00:09:41.548 So that's going to be 1, 7 plus 2 7th he calls 3. 70 00:09:43.229 --> 00:09:49.499 And so in total probability therapist, and so on. 71 00:09:49.499 --> 00:09:54.658 Okay, so again, this is, um. 72 00:09:58.168 --> 00:10:01.979 This let me write this down, so people can understand it better. 73 00:10:03.928 --> 00:10:13.769 This is a fictional University question. 74 00:10:13.769 --> 00:10:20.969 Okay, any questions about that. 75 00:10:22.739 --> 00:10:27.448 So base. 76 00:10:30.899 --> 00:10:34.318 On here. 77 00:10:39.658 --> 00:10:47.099 Okay, good. Um, Wikipedia. 78 00:10:47.099 --> 00:10:54.538 People complain about Wikipedia a lot, but I find that for technical questions. It is actually. 79 00:10:54.538 --> 00:10:59.038 Most of the time quite good. So. 80 00:10:59.038 --> 00:11:05.068 I also like to give you different ways to see the same thing. 81 00:11:05.068 --> 00:11:11.038 So, you see how I talk about it you see our professor read key talks about it. 82 00:11:11.038 --> 00:11:20.489 You see how the book talks about a PDF and you see all these different ways of presenting the same topic. 83 00:11:20.489 --> 00:11:26.278 Perhaps you might start understanding it and so, Wikipedia. 84 00:11:26.278 --> 00:11:30.869 On base serum isn't bad. 85 00:11:32.639 --> 00:11:46.349 So, what I'm going to do is use an example here. Oh, 1, thing, tying into the real world and vaccine since they're in the notice, now there's concepts to specify specificity. 86 00:11:46.349 --> 00:11:52.619 And sensitivity, and we'll talk about a more later just ways to talk about how good a test is. 87 00:11:54.028 --> 00:12:03.448 Okay, I'm going to run through this example right here actually. So this is. 88 00:12:13.948 --> 00:12:17.759 Okay. 89 00:12:17.759 --> 00:12:26.818 Cool. So, and also, these examples of courses actually, you know, examples that ties to the real world. 90 00:12:26.818 --> 00:12:30.089 I'd like why I like this course and. 91 00:12:30.089 --> 00:12:36.509 You have a machine here, and I'm not also not going to hand write everything that. 92 00:12:36.509 --> 00:12:41.698 Try to send right extra things. We have a, we have a factory. 93 00:12:41.698 --> 00:12:47.668 And it's producing some widget, and it's got 3 different machines that produce the widget. 94 00:12:47.668 --> 00:12:51.448 Machine a. B and C. 95 00:12:51.448 --> 00:12:57.869 Now, these machines have different production rates, different productivity. 96 00:12:57.869 --> 00:13:02.729 Um, machine a produces. 97 00:13:02.729 --> 00:13:11.879 20% of their total widgets machine B produces 30% and machine sees the fastest produces 50 per cent. 98 00:13:11.879 --> 00:13:20.188 So, I'm thinking of your Tesla Motors, for example, and they're getting batteries some batteries and Panasonic, some batteries they make himself. 99 00:13:20.188 --> 00:13:26.729 Would be an example, or Ivan an inverters say I just got. 100 00:13:26.729 --> 00:13:30.328 Solar panels on my house solar panels. 101 00:13:30.328 --> 00:13:37.589 They have a, um, an inverter from 1 company not Tesla. 102 00:13:37.589 --> 00:13:45.869 Not the best reputation, perhaps some tesla's moved over to producing their own inverters to make any case. So, this example, here. 103 00:13:45.869 --> 00:13:56.339 We have the widget 3 machines are producing the widget a B and C different speeds. The machines also have different reliabilities. 104 00:13:56.339 --> 00:14:04.019 Machine a of the widgets that machine a produces 95% are good and 5% are bad. 105 00:14:04.019 --> 00:14:10.168 Machine B is better. 97% are good and only 3% or bad. 106 00:14:10.168 --> 00:14:13.288 Machine C. is the best. 107 00:14:13.288 --> 00:14:19.078 99% of machines seas. Widgets are good and 1% are bad. 108 00:14:19.078 --> 00:14:23.278 So machine C, it's fast and it's also high quality. 109 00:14:25.558 --> 00:14:39.389 So, the question is that these machines that we're just all being producer, all jumbled together in the band, or something you pull a random widget out of the band. It's a 1000 widgets in the band all mixed up randomly. 110 00:14:39.389 --> 00:14:44.369 You pull 1 widget out of the bin and then the question is. 111 00:14:45.389 --> 00:14:49.349 And it's a bandwidth. Well, 1st question is. 112 00:14:50.849 --> 00:14:56.249 What's the probability that the widget is good or what's the probability that it's bad? 113 00:14:56.249 --> 00:15:01.048 And then the 2nd question is that if the widget is bad. 114 00:15:01.048 --> 00:15:05.759 What's the probability that it was made by machine? C. 115 00:15:07.558 --> 00:15:19.349 Okay, now there's 2 ways you can solve this question at least. Well, it's really easy. 1 if we just do a table, like, we've got up there. 116 00:15:20.519 --> 00:15:29.428 And a number of widgets produced by machines a B and C or assuming it's a 1000 widgets in total. 117 00:15:29.428 --> 00:15:34.528 Machine C produces 50% so machine see produces 500. 118 00:15:34.528 --> 00:15:43.198 Be 30%. That's 320%. That's 200. so, that's the total thing that's number, which is produced by each machine. 119 00:15:43.198 --> 00:15:47.068 Then we have the reliability numbers, so machine a. 120 00:15:47.068 --> 00:15:53.068 Has 5% defective so it produces 200 widgets of those 200 widgets. 121 00:15:53.068 --> 00:16:00.778 5% is 10. that's 10 bad ones. B3% of its 300 widgets are bad. That makes 9. 122 00:16:00.778 --> 00:16:07.438 See, 1% of it's 500. we're just so bad that makes 5. that's the defective column there. 123 00:16:07.438 --> 00:16:11.249 Okay, now we add them up and there's. 124 00:16:11.249 --> 00:16:15.448 24 bad widgets of the 1000, so. 125 00:16:15.448 --> 00:16:18.479 In total there's 2.4. 126 00:16:18.479 --> 00:16:22.349 Percent bad widgets. 127 00:16:22.349 --> 00:16:29.729 And then the base Beijing question would be if we get 1 of those 24 bad widgets. 128 00:16:29.729 --> 00:16:33.389 What's the probability that it was made by machine? C. 129 00:16:33.389 --> 00:16:37.979 Well, of those 24 bad widgets, we can look in the table. 130 00:16:37.979 --> 00:16:43.918 And seen that 5 of them are from machine C. so, 524 are. 131 00:16:45.568 --> 00:16:49.708 So so 520 force of the bad widgets. 132 00:16:49.708 --> 00:16:54.298 Are from machine C. so that's just a simple counting argument. 133 00:16:54.298 --> 00:16:58.708 But I want to show you using our formulas because. 134 00:16:59.818 --> 00:17:03.899 The formulas are a more general solution. Okay. 135 00:17:03.899 --> 00:17:11.848 So, let's write down what we have here. 136 00:17:14.189 --> 00:17:19.259 So, a means that widget a random widget was produced. 137 00:17:19.259 --> 00:17:24.179 By machine a, and that equals point 0, 2. 138 00:17:24.179 --> 00:17:32.519 Sorry point to machine B. 139 00:17:32.519 --> 00:17:38.519 Produce 30% point 3 5. 140 00:17:38.519 --> 00:17:46.108 And then we're also told how good each machine is. So the probability. So D will be. 141 00:17:48.088 --> 00:17:52.828 Defective is. 142 00:17:52.828 --> 00:17:58.259 Silence. 143 00:17:58.259 --> 00:18:05.159 Again, the vertical bar means given the probability that it's a defective, which had given that it's. 144 00:18:05.159 --> 00:18:09.778 And I, widget came out of machine a is point 0, 5. 145 00:18:11.729 --> 00:18:16.259 Effective given it came out of machine B is, um. 146 00:18:17.368 --> 00:18:23.699 3 effective get gave see is point on 1. 147 00:18:23.699 --> 00:18:36.509 Okay, so now we can add up and now we can do the probability of it being from machine a, and also defective. We multiply. 148 00:18:37.769 --> 00:18:41.159 And what does the right down the formula. 149 00:18:47.489 --> 00:18:54.598 Equals, um. 150 00:18:55.739 --> 00:18:59.308 Okay, um, the probability. 151 00:18:59.308 --> 00:19:03.358 Of the, and the fact of to be there. 152 00:19:03.358 --> 00:19:08.548 He calls. 153 00:19:08.548 --> 00:19:12.868 Equals. 154 00:19:16.679 --> 00:19:24.959 Okay, so 2nd, here. Oh, 9. 155 00:19:24.959 --> 00:19:34.499 And, um, oh, really see and effective. 156 00:19:34.499 --> 00:19:41.308 Is point. 157 00:19:42.358 --> 00:19:47.308 And so the problem of being defective is to some of those 3. 158 00:19:56.669 --> 00:20:03.269 Equals, um, okay. 159 00:20:05.788 --> 00:20:10.588 Now, we want to go and so now what we want. 160 00:20:10.588 --> 00:20:15.659 Is a probability of C given that it was the effective. 161 00:20:15.659 --> 00:20:24.328 Well, see and effective. 162 00:20:24.328 --> 00:20:34.019 Over probability of defective probably to see and defective was point all 5 probability of being 2 factors point or 2 or. 163 00:20:34.019 --> 00:20:37.348 Of course. 164 00:20:37.348 --> 00:20:42.989 Okay, so that is. 165 00:20:42.989 --> 00:20:46.769 A nice example of Asian reasoning. You see, we're going. 166 00:20:46.769 --> 00:20:57.689 Backwards to the forward way is given its come from machine. See, what's probably it's effective we have that we want to go the other way of given that it's effective. 167 00:20:57.689 --> 00:21:06.509 What's the probability it came from machine? See, if you've got any questions on mute your Mike or. 168 00:21:06.509 --> 00:21:10.288 Is it just a coincidence that it looks like. 169 00:21:10.288 --> 00:21:15.808 It says 24, it's that it's. 170 00:21:15.808 --> 00:21:23.999 Is it a coincidence that well, it's the same number. It's the same answer I got from the counting argument by just. 171 00:21:23.999 --> 00:21:29.219 Counting down here yet I was doing the same problem 2 different ways. So. 172 00:21:29.219 --> 00:21:33.269 I've got to get the same answer. Is that what you meant? 173 00:21:35.429 --> 00:21:40.259 Yeah, okay. So. 174 00:21:40.259 --> 00:21:45.868 Uh, when you say that notation and intersect years, the safety. 175 00:21:45.868 --> 00:21:50.788 What does that does that also mean? You said it also means empty like. 176 00:21:50.788 --> 00:21:55.858 Oh, okay. Notation. 177 00:21:55.858 --> 00:21:59.788 Okay. 178 00:22:01.648 --> 00:22:08.909 So come on. Okay, so. 179 00:22:10.019 --> 00:22:16.169 But this means this means that's the probability. 180 00:22:21.898 --> 00:22:31.259 What did you. 181 00:22:31.259 --> 00:22:34.499 And also. 182 00:22:34.499 --> 00:22:38.338 Is the factor. 183 00:22:40.078 --> 00:22:47.909 Okay, so with this thing here means. 184 00:22:52.709 --> 00:22:59.429 This mean, this is an intersect. 185 00:23:02.909 --> 00:23:11.519 Silence. 186 00:23:12.568 --> 00:23:16.919 Okay. 187 00:23:16.919 --> 00:23:20.398 Oh, yeah, yeah. Okay. 188 00:23:22.858 --> 00:23:27.598 Silence. 189 00:23:29.128 --> 00:23:38.608 Okay, we get the same thing also another example is say. 190 00:23:39.959 --> 00:23:44.159 Um. 191 00:23:47.189 --> 00:23:51.269 Even a, it's even. 192 00:23:53.459 --> 00:23:59.219 A prime or something. 193 00:23:59.219 --> 00:24:02.909 So, what we would have. 194 00:24:04.919 --> 00:24:09.058 So, a, is going to be the set of 2 4. 195 00:24:09.058 --> 00:24:18.808 6 B is going to be the set of 2 and 3 and 5, for example. So we could talk about. 196 00:24:18.808 --> 00:24:22.769 A intersect B is going to be the set of 2. 197 00:24:22.769 --> 00:24:28.439 For example, and we could we could do this also that, um. 198 00:24:28.439 --> 00:24:33.088 Fair dye. Probably today is 1 half. 199 00:24:35.459 --> 00:24:42.778 It goes 1, half probability of a intersex B is going to be 1 6. 200 00:24:44.848 --> 00:24:49.919 Probability of that it's even given that it's Prime. 201 00:24:55.138 --> 00:25:01.169 Equals. 202 00:25:01.169 --> 00:25:04.828 Very example. 203 00:25:05.848 --> 00:25:09.808 It's also example of conditional probability. 204 00:25:09.808 --> 00:25:13.199 And again, just for a re. 205 00:25:13.199 --> 00:25:18.118 Refresh what this means equals the probability. 206 00:25:19.709 --> 00:25:24.028 Of a. 207 00:25:24.028 --> 00:25:28.048 Be happened. 208 00:25:31.679 --> 00:25:35.128 So, in this case here we toss the dye. 209 00:25:35.128 --> 00:25:38.818 If we. 210 00:25:38.818 --> 00:25:42.598 Saw and prime number. 211 00:25:42.598 --> 00:25:46.439 What's the probability that it's also even, for example. 212 00:25:46.439 --> 00:25:49.979 Okay. 213 00:25:53.548 --> 00:25:56.759 Survey serum. 214 00:25:56.759 --> 00:26:00.058 Want to gave you some other examples. 215 00:26:00.058 --> 00:26:03.538 And I get this a little last time, but I'll. 216 00:26:03.538 --> 00:26:07.798 Repeat them again slightly differently perhaps just so that. 217 00:26:07.798 --> 00:26:12.179 People can see them I did the disease thing. I want to the disease thing again. 218 00:26:12.179 --> 00:26:17.038 But let's say page 51, it's a book. 219 00:26:39.328 --> 00:26:46.229 Well, I'll just walk you through it somewhat. 220 00:26:50.249 --> 00:26:53.578 It's like the Wikipedia example. 221 00:26:54.838 --> 00:27:02.848 But here, we're not just producing a fixed number. We're continuously producing and they're either good chips or bad chips and. 222 00:27:04.588 --> 00:27:16.318 The good chips, both chips are failing with an exponential probability distribution, which we'll get to later. 223 00:27:16.318 --> 00:27:20.729 Silence. 224 00:27:20.729 --> 00:27:27.778 Okay. 225 00:27:31.048 --> 00:27:36.719 Ok, so so the good chip is so the probability. 226 00:27:36.719 --> 00:27:40.138 That so the way this is written. 227 00:27:45.838 --> 00:27:50.999 It's an alpha in there. So what this means here is, this is the probability. 228 00:27:53.638 --> 00:27:59.669 A good. 229 00:27:59.669 --> 00:28:06.058 Is still alive at time T. 230 00:28:06.058 --> 00:28:10.108 It's going to plot something like this. 231 00:28:10.108 --> 00:28:17.878 Time when someone goes down to 0T, it's decreasing that. They're right. Just more neatly. 232 00:28:22.259 --> 00:28:25.618 Minus alpha. 233 00:28:25.618 --> 00:28:29.548 God oh, okay. That's the good chip. 234 00:28:29.548 --> 00:28:34.378 The bad chips are going to go something like this. 235 00:28:37.259 --> 00:28:43.108 So that's bad trip. This is good. Chip. 236 00:28:44.398 --> 00:28:48.088 Okay, okay so. 237 00:28:49.739 --> 00:28:54.209 And there's a certain fraction of the chips are good or bad and. 238 00:28:54.209 --> 00:29:03.358 What we see there what I'm showing on a display from the, from the book is the probability that. 239 00:29:03.358 --> 00:29:07.259 A random chip is still good at time tea. 240 00:29:08.308 --> 00:29:12.118 Okay, and we don't know which it is. 241 00:29:12.118 --> 00:29:17.338 But, okay, so now the way you would apply base theorem to this. 242 00:29:17.338 --> 00:29:24.269 Is 2 things you can do? Yeah, I'll leave it on here. 243 00:29:24.269 --> 00:29:29.878 Well, 1st, the well question if a chip is good. 244 00:29:31.409 --> 00:29:35.368 Then, what's the if the chip is still alive. 245 00:29:35.368 --> 00:29:39.628 What's the probability that it's a good ship? That it was a good ship? 246 00:29:39.628 --> 00:29:43.618 And the longer we burn them in. 247 00:29:43.618 --> 00:29:48.689 The higher the percentage of the of the surviving chips are that are good. 248 00:29:48.689 --> 00:29:51.719 So then the question would be. 249 00:29:51.719 --> 00:29:59.128 Um, to spend knows how long would we have to burn them in? So it's a 99% chance. 250 00:29:59.128 --> 00:30:10.648 That the chips are good at a random chip is good. Let me write some of this, this anticipating a little but since we're talking about this now. 251 00:30:11.784 --> 00:30:12.354 Okay, 252 00:30:12.354 --> 00:30:35.874 silence. 253 00:30:37.828 --> 00:30:43.348 Portion. 254 00:30:53.548 --> 00:30:57.118 Or good. 255 00:30:58.949 --> 00:31:06.509 It goes bad chips, die, faster, sooner. 256 00:31:08.844 --> 00:31:23.544 The big question batch. 257 00:31:23.939 --> 00:31:27.929 So that. 258 00:31:32.788 --> 00:31:41.909 Silence. 259 00:31:41.909 --> 00:31:45.959 Professor yes, I ask a question about what you mean by. 260 00:31:45.959 --> 00:31:52.979 In batch well, we manufactured a pile of chips. 261 00:31:52.979 --> 00:31:57.088 And some of them. 262 00:31:57.088 --> 00:32:05.038 We're done, right and some of them something screwed up in the process so they work, but they're going to die fast. 263 00:32:05.038 --> 00:32:12.028 And I'm calling them the good chips and the bad chips of the bad chips still function. 264 00:32:12.028 --> 00:32:15.209 Currently, but they're going to die sooner. 265 00:32:16.618 --> 00:32:20.398 And we have this exponential distribution. 266 00:32:20.398 --> 00:32:27.028 For how for the probability that either good chip or a bad ship is still alive and time tea. 267 00:32:27.028 --> 00:32:30.929 So, we got this, we made this, um, box full of chips. 268 00:32:30.929 --> 00:32:36.239 Some good some bad and we burn them all in. We're putting a minute we power them up. 269 00:32:36.239 --> 00:32:43.469 And we run them, and in that box of chips, the bad ones are going to die faster. 270 00:32:43.469 --> 00:32:48.328 So, of those chips that are still alive at time tea. 271 00:32:48.328 --> 00:32:55.169 A larger as time goes on a larger percentage will be the good chips because the good chips live longer. 272 00:32:55.169 --> 00:33:03.838 And the question is, how long do we have to burn the whole box of chips in. 273 00:33:03.838 --> 00:33:11.128 So that the ones that are still alive, have a 99% chance of being the good chips, which means they're going to continue to last a long time. 274 00:33:11.128 --> 00:33:14.278 Okay, and thank you sure. 275 00:33:14.278 --> 00:33:17.519 I have a question that's kind of unrelated. 276 00:33:17.519 --> 00:33:23.939 Ask away, how would they determine that it follows an exponential law? 277 00:33:23.939 --> 00:33:29.818 Compete or something, or because it made the example and nice example. 278 00:33:30.838 --> 00:33:36.568 In the real world it may, or may not be true. It depends on what's causing the chips to die. 279 00:33:38.459 --> 00:33:46.259 So, if they're dying, because they're getting hit by cosmic rays or radioactivity. 280 00:33:46.259 --> 00:33:50.219 Steel. 281 00:33:50.219 --> 00:33:54.989 For example, steel, today's a small amount of radiation and it as does concrete. 282 00:33:54.989 --> 00:33:59.699 So, if that's what's killing it, then if not. 283 00:33:59.699 --> 00:34:05.368 A real world failure a thing they mentioned later in the book. 284 00:34:16.139 --> 00:34:23.099 Perhaps, um, time. 285 00:34:27.119 --> 00:34:32.789 Probably are dying there might be something like this. 286 00:34:32.789 --> 00:34:35.998 Here we have sort of initial. 287 00:34:37.918 --> 00:34:41.009 You know, whatever, worst defects. 288 00:34:42.298 --> 00:34:47.699 And here we have old age or something. 289 00:34:47.699 --> 00:34:51.748 And here we have yeah, so. 290 00:34:51.748 --> 00:34:56.909 You said, they just picked the example to make it work. Nice. 291 00:34:56.909 --> 00:35:01.199 Okay, so. 292 00:35:03.898 --> 00:35:08.128 So that's the chip quality control here thing. 293 00:35:08.128 --> 00:35:11.548 Okay, um. 294 00:35:17.278 --> 00:35:25.048 Another example, here, the binary communications thing, I did it again. I did it last time. I'll just say this. 295 00:35:25.048 --> 00:35:29.099 Um, and 230 is what I was just working. 296 00:35:29.099 --> 00:35:32.789 2009 here. 297 00:35:32.789 --> 00:35:42.688 Again, you're transmitting, say zeros or ones, and maybe it's an image of a page so there's more 0 0T means white. 1 means black. Maybe there's more zeros and ones the transmission. 298 00:35:42.688 --> 00:35:50.969 Has a small probability of going wrong if you receive a 0T or 1 then what's the probability that that was transmitted? 299 00:35:52.378 --> 00:35:55.528 And if the error is large enough, then. 300 00:35:55.528 --> 00:36:01.079 That most probably will receive thing might be not what was transmitted if the air is so large. 301 00:36:01.079 --> 00:36:15.088 Well, the example I used last time was werewolf, as let's say, if there's a small percentage of the population is aware, and you have a test for it. But the test is a fairly large error. 302 00:36:15.088 --> 00:36:24.298 Even if you're positive, it doesn't mean you're aware off, it raises the probability that you're aware. Well, but it may not raise it as far as the half. 303 00:36:24.298 --> 00:36:28.139 Quality control this thing here, so. 304 00:36:28.139 --> 00:36:35.039 And that's how I'm not going to write it down in the complicated enough independence of events. Okay. 305 00:36:37.079 --> 00:36:40.949 This means that if you make. 306 00:36:40.949 --> 00:36:44.579 2 observations on the same experiment. 307 00:36:44.579 --> 00:36:49.648 And does what you learn from the 1st, 1 change the probability. 308 00:36:49.648 --> 00:36:55.498 Of the 2nd, 1 occurring and is a formal definition, right there. 309 00:36:55.498 --> 00:36:59.068 And of independent. 310 00:36:59.068 --> 00:37:04.108 And if you got more than 2 events you have to do in all combinations. 311 00:37:05.878 --> 00:37:10.648 So, I talk about independent events and and so on the the. 312 00:37:10.648 --> 00:37:17.009 So, probably, they given B is equal to the probability of a given. B doesn't change anything. 313 00:37:18.449 --> 00:37:21.898 It's a definition, but it matches your intuition and someone. 314 00:37:21.898 --> 00:37:30.148 But I'd like to do is just go through some examples and since you can read the books typing better than you can read my hand printing, though. 315 00:37:30.148 --> 00:37:34.048 Box for them. 316 00:37:35.159 --> 00:37:39.088 Skip that 1, I mean, we can get as much. 317 00:37:43.108 --> 00:37:46.798 Okay, and I'll sketch it down here. 318 00:37:46.798 --> 00:37:52.739 Also. 319 00:37:57.869 --> 00:38:03.599 Page, uh, 55 give her. 320 00:38:03.599 --> 00:38:08.818 Okay, so we're picking 2 numbers. 321 00:38:08.818 --> 00:38:16.139 And and the problem, we can have events, you know, you can define. 322 00:38:16.139 --> 00:38:19.139 Events to be anything you want. 323 00:38:19.139 --> 00:38:25.500 And the outcomes are what X and Y, are than the events are more complicated things. 324 00:38:25.500 --> 00:38:31.380 So, for example, event a, is X is greater than a half. 325 00:38:31.380 --> 00:38:34.409 Event B, is why is greater than a half. 326 00:38:34.409 --> 00:38:37.769 And event sea is X is greater than why. 327 00:38:39.179 --> 00:38:42.989 And you could even plot it actually. 328 00:38:50.039 --> 00:38:55.860 And so a X, greater than a half let me pick colors here. 329 00:39:00.239 --> 00:39:04.619 It's greater than a half doesn't matter what why is. 330 00:39:04.619 --> 00:39:09.719 Event B, is why is greater than a half. 331 00:39:11.730 --> 00:39:19.739 And event see, is X is greater than why. 332 00:39:19.739 --> 00:39:23.730 Just get some new colors here. 333 00:39:26.400 --> 00:39:32.610 X is greater than why. 334 00:39:32.610 --> 00:39:37.679 So is going to be up here. 335 00:39:39.719 --> 00:39:43.559 That would be events. See okay. 336 00:39:45.659 --> 00:39:48.719 Now, we can do probabilities here. 337 00:39:51.059 --> 00:39:57.090 Um, good. 338 00:39:58.469 --> 00:40:02.010 Trying to get at to show both of them simultaneously. 339 00:40:02.010 --> 00:40:09.719 It's not working, so well, probably today is 1 half. I probably will be as 1 half file so. 340 00:40:11.369 --> 00:40:22.590 It was probably LTC. How do we know that CX is greater than why? When we look in this figure here it's an upper triangle. It's above that diagonal line. Okay. 341 00:40:23.639 --> 00:40:27.000 Now, we can start doing things like. 342 00:40:28.590 --> 00:40:35.460 Well, we can look at probability of X greater than half and why greater than half. 343 00:40:35.460 --> 00:40:38.460 That will be a intersect B. 344 00:40:38.460 --> 00:40:43.710 If we look at the diagram here for that X, greater than a half. 345 00:40:43.710 --> 00:40:49.679 Is here migrated and a half is here and the. 346 00:40:53.610 --> 00:40:58.349 The both of them is here. Okay. 347 00:41:00.239 --> 00:41:10.139 And that's going to be 1 quarter, and that's also equal to the probability of a time. So. 348 00:41:10.139 --> 00:41:19.829 So, they're independent now, if I look at the probabilities say. 349 00:41:21.300 --> 00:41:30.360 Of a, and C, so a, was X greater than 1 half and C was greater than. 350 00:41:30.360 --> 00:41:34.079 Why, well, if we plot that, um. 351 00:41:35.699 --> 00:41:39.059 X, greater than a half is up here. 352 00:41:39.059 --> 00:41:45.780 X greater than Y is here and the, both of them together. 353 00:41:45.780 --> 00:41:52.320 Are here and that's not a quarter. That's an 8 that chilly. 354 00:41:55.710 --> 00:42:05.429 And that does not equal. So what this means in English. 355 00:42:05.429 --> 00:42:10.860 Is that the probability? Let me actually. 356 00:42:13.440 --> 00:42:16.739 No, I want in the next. I'm sorry. 357 00:42:16.739 --> 00:42:19.769 Well, you said it is no migrate or anything that. 358 00:42:21.630 --> 00:42:28.349 Uh, could well be yeah, you're right. 359 00:42:37.170 --> 00:42:40.980 Snyder again, that's a worthwhile point. Send me an email. 360 00:42:40.980 --> 00:42:47.460 Okay, X, greater than a half. 361 00:42:47.460 --> 00:42:51.719 Is here. 362 00:42:52.860 --> 00:42:56.730 X greater than Y, here. 363 00:42:59.789 --> 00:43:05.639 And so the 2 of them together are this thing here. 364 00:43:08.760 --> 00:43:12.449 Chile, which is. 365 00:43:14.639 --> 00:43:17.909 And 5 days. 366 00:43:20.519 --> 00:43:23.579 So, in English is that, um. 367 00:43:23.579 --> 00:43:27.210 So, they're not independent. 368 00:43:30.420 --> 00:43:36.420 So, an English, what that means. 369 00:43:36.420 --> 00:43:40.170 If X is greater than 1, half. 370 00:43:45.659 --> 00:43:49.889 It's more likely that X is greater than why. 371 00:43:53.460 --> 00:44:03.599 Then if we know nothing about X okay. 372 00:44:03.599 --> 00:44:10.409 So, and that we could do the conditional. 373 00:44:12.090 --> 00:44:20.130 Probability of see, given a probability of a and C divided by a probability of a. 374 00:44:22.170 --> 00:44:27.780 Which is it close? 375 00:44:31.469 --> 00:44:35.280 It's not 5, 8, it's 5. sixteenths. 376 00:44:50.400 --> 00:44:54.750 Yeah, so, um. 377 00:44:56.159 --> 00:44:59.909 It got more and this, right? 378 00:45:05.730 --> 00:45:17.400 No, 3. 379 00:45:18.599 --> 00:45:29.010 Along sorry, everybody 1 half equals 3 quarters. 380 00:45:30.420 --> 00:45:39.960 That makes sense. Okay. So the probability that X is greater than half and is also greater than why is. 381 00:45:39.960 --> 00:45:43.590 3, 8, it's less than the probably X greater than. 382 00:45:43.590 --> 00:45:50.429 Why, and so they probably, if X greater than a, why give it a greater than a half is 3 quarters. 383 00:45:50.429 --> 00:45:56.190 No dependent events. 384 00:45:56.190 --> 00:45:59.369 Yellow areas 3 eighths yes. 385 00:45:59.369 --> 00:46:04.260 Okay. 386 00:46:04.260 --> 00:46:10.889 And and they're working their way through here somewhat. 387 00:46:14.280 --> 00:46:18.300 They have another example here. 388 00:46:24.269 --> 00:46:30.329 Okay, 233 raises a new point. 389 00:46:38.190 --> 00:46:46.800 Silence. 390 00:47:01.559 --> 00:47:20.280 Silence. 391 00:47:25.320 --> 00:47:29.429 And this gets messy and the example here. 392 00:47:29.429 --> 00:47:34.019 Is that we've got 3 events here. 393 00:47:34.019 --> 00:47:37.920 They call them B. D. and f. 394 00:47:37.920 --> 00:47:42.030 And the India are independent. 395 00:47:43.440 --> 00:47:48.570 For independent DNS are independent. 396 00:47:49.889 --> 00:47:55.739 Are not independent. 397 00:47:57.300 --> 00:48:02.820 And what that means is that the definition is that the probability. 398 00:48:02.820 --> 00:48:06.659 Of the anti. 399 00:48:06.659 --> 00:48:10.650 And f, it's not equal to the probability of the. 400 00:48:14.099 --> 00:48:19.260 And they have the example. 401 00:48:20.969 --> 00:48:25.320 Throwing the numbers in. 402 00:48:31.980 --> 00:48:35.159 So B, is why is greater than a half. 403 00:48:35.159 --> 00:48:41.460 Um. 404 00:48:42.599 --> 00:48:47.039 D, is excess less than a half. 405 00:48:55.889 --> 00:49:00.420 And. 406 00:49:00.420 --> 00:49:08.849 If it is a complicated sort of thing, I'm just going to draw separately. 407 00:49:10.980 --> 00:49:15.179 F, is they're both less than a half. 408 00:49:16.559 --> 00:49:20.610 Or they're both greater than a half. That's half. 409 00:49:20.610 --> 00:49:24.030 And. 410 00:49:26.579 --> 00:49:32.010 Silence. 411 00:49:33.420 --> 00:49:38.460 I can't actually showed 2 pages together. It's sort of annoying. 412 00:49:39.539 --> 00:49:44.670 But the probability 1, half, the problem is. 413 00:49:44.670 --> 00:49:49.079 Is 1 half probably to B and D. 414 00:49:49.079 --> 00:49:56.820 Equals 1 quarter if I just look at this thing down here, that's a double colored thing. So they're independent. 415 00:49:59.519 --> 00:50:03.989 But if I look at B and f together. 416 00:50:03.989 --> 00:50:12.630 They're independent because bnf, he'll be the top right quadrant and that will still be 1 quarter. 417 00:50:14.070 --> 00:50:23.280 Um, that's the top right corner. 418 00:50:27.659 --> 00:50:32.070 This equals 1 quarter is a phone would be. 419 00:50:32.070 --> 00:50:35.940 Probably have half independent. 420 00:50:37.230 --> 00:50:42.780 They however all 3 together. 421 00:50:45.119 --> 00:50:50.519 Um, the N. D and f0. 422 00:50:50.519 --> 00:50:57.329 And go back up here you see all 3 together, it never happens. 423 00:50:57.329 --> 00:51:00.929 Um, so. 424 00:51:00.929 --> 00:51:06.329 So the 3 considered as a triple are dependent. 425 00:51:06.329 --> 00:51:11.039 If we know 2 of them, what this means in English. 426 00:51:11.039 --> 00:51:17.070 Is that if we know 2 of them, this tells us something about the 3rd. 427 00:51:18.480 --> 00:51:23.010 Write this down, so. 428 00:51:25.619 --> 00:51:29.489 2. 429 00:51:31.530 --> 00:51:35.250 Of them occurred. 430 00:51:35.250 --> 00:51:38.940 Silence. 431 00:51:38.940 --> 00:51:44.550 And we know something about the 3rd. 432 00:51:44.550 --> 00:51:50.820 The 3rd event. 433 00:51:52.769 --> 00:51:55.769 Here. 434 00:51:55.769 --> 00:52:02.969 Post B and D are true. 435 00:52:06.480 --> 00:52:10.320 Then, if is impossible. 436 00:52:13.469 --> 00:52:16.769 So dependent. 437 00:52:19.199 --> 00:52:25.650 Dependent. 438 00:52:29.219 --> 00:52:35.969 Okay, I mean, the real world things get complicated. 439 00:52:35.969 --> 00:52:42.119 So, and then here you could get it. 440 00:52:42.119 --> 00:52:50.670 Any combination and events being dependent and so on you're talking about so I set the way. 441 00:52:50.670 --> 00:52:56.730 If I was in the bottom. 442 00:52:59.099 --> 00:53:04.139 Uh, no, that's just a compliment of the other case. 443 00:53:04.139 --> 00:53:11.070 If half independent of what of 1 of the others or both of the others. 444 00:53:11.070 --> 00:53:19.559 Well, it would be independent of B, wouldn't be independent of day, but it would still be dependent on both. B and D. all 3 would still be dependent. 445 00:53:22.170 --> 00:53:25.889 Um, I think. 446 00:53:25.889 --> 00:53:29.369 Okay, maybe you're right I'm wrong now. Let me see here. 447 00:53:31.710 --> 00:53:35.070 Okay, let's look at this here. 448 00:53:35.070 --> 00:53:39.150 Okay. 449 00:53:39.150 --> 00:53:46.800 So, it's about okay, so the question. 450 00:53:46.800 --> 00:53:53.489 Well, let's look at that. Um, so if B. 451 00:53:53.489 --> 00:53:58.170 Was up here and then D was. 452 00:53:58.170 --> 00:54:02.309 Here now we're getting this. 453 00:54:02.309 --> 00:54:06.119 And now we're getting so, what results is here. 454 00:54:07.380 --> 00:54:11.159 And then we add in and we add in your new app. 455 00:54:11.159 --> 00:54:16.500 It's here and here and it results at this here. 456 00:54:18.570 --> 00:54:25.170 Um, no. 457 00:54:28.289 --> 00:54:37.949 Let's call this g. let's say it's g, no, because the probability of G. 458 00:54:37.949 --> 00:54:44.699 Is 1 half the probability of B and D. N. G. 459 00:54:44.699 --> 00:54:49.230 Would be 1 quarter and that's not equal to. 460 00:54:49.230 --> 00:54:52.949 1, half 1, half and half so they still. 461 00:54:52.949 --> 00:54:57.780 They're still dependent, so. 462 00:55:01.920 --> 00:55:06.239 The triple is still dependent. Okay. 463 00:55:08.880 --> 00:55:15.389 That makes sense because if there's independence, if you just compliment the variable. 464 00:55:15.389 --> 00:55:20.849 It doesn't change it enough. It's still going to be dependent. 465 00:55:23.369 --> 00:55:26.880 Okay, okay. 466 00:55:30.570 --> 00:55:40.530 Now, coins, we assume successive tosses are independent of each other. If we have no reason not to. 467 00:55:41.760 --> 00:55:48.719 If it was fair coin side. 468 00:55:48.719 --> 00:55:52.440 So, that's what they're talking about here. 469 00:55:54.059 --> 00:55:59.190 Reliability this is a nice example. I wrote this up on the blog a little. 470 00:56:01.829 --> 00:56:05.880 Not as complicated as that burning curve I showed you, but. 471 00:56:07.110 --> 00:56:14.010 What's happening here is that the system's got a controller and 3 peripheral units. 472 00:56:14.010 --> 00:56:26.489 And you need, at least for the system to function, you need to controller, and at least 2 of the peripherals to operate, if 0T or 1 of the peripherals are operating in the whole system is dead. 473 00:56:26.489 --> 00:56:32.880 If 2 or 3 of the are running, the system is good. If the controller, the controller is also running. 474 00:56:34.289 --> 00:56:37.889 And we can start calculating probabilities here. 475 00:56:37.889 --> 00:56:44.789 Um, and. 476 00:56:53.340 --> 00:56:57.179 Raise the probability of peripheral fails and. 477 00:56:57.179 --> 00:57:00.929 So this is the probability that. 478 00:57:00.929 --> 00:57:08.429 Here that 2 or 3 other peripherals failed. So this is a probability at all. 3 failed. 479 00:57:08.429 --> 00:57:13.349 This is the probability of the 2 or 3 failed could be any and the 1 that failed. 480 00:57:13.349 --> 00:57:19.619 The 2 that failed to be any 2 of the 3. this is a combinate question here. 481 00:57:19.619 --> 00:57:28.380 And then we have to the whole thing that we have to bring in, that the controller failure probably and we end up down here. 482 00:57:29.610 --> 00:57:32.820 The probability that the system is up so. 483 00:57:34.110 --> 00:57:37.800 What this means this is that the problem, but it's controller did not fail. 484 00:57:37.800 --> 00:57:41.429 And this is the probability that at least 2 peripherals are still working. 485 00:57:41.429 --> 00:57:48.449 And we're assuming their independent so. 486 00:57:50.730 --> 00:57:55.889 Sequential experiments are tossing a coin 10 times or something. 487 00:57:55.889 --> 00:58:05.130 And we usually assume that they're independent here that the subsystems are independent. So. 488 00:58:05.130 --> 00:58:09.090 Unless we've got some reason not to assume that. 489 00:58:12.599 --> 00:58:16.409 And what would if I type here on the blog. 490 00:58:17.579 --> 00:58:24.420 We've got it ran into this. This is the thing this is just writing down in words what? I just wrote down by hand. 491 00:58:24.420 --> 00:58:29.610 As I said, coin tosses, we assume that they are independent of each other. 492 00:58:29.610 --> 00:58:35.429 Unless we know otherwise. 493 00:58:37.440 --> 00:58:46.019 Sequential experiments, we assume they're independent unless we've got a reason not to do that. Who knows what that might be so. 494 00:58:48.929 --> 00:58:58.469 Yeah, okay well this is called again. Good chance to review it. 495 00:58:58.469 --> 00:59:05.280 Well, just here's the order matters somewhat so. 496 00:59:05.280 --> 00:59:10.559 If we're picking 10 numbers, and we want the 1st, 4 to be less than a quarter. 497 00:59:10.559 --> 00:59:15.780 And the last 1st, 5, I'm sorry in the last 5 to be greater than a half. 498 00:59:17.039 --> 00:59:21.599 Well, then this is nice and easy, and we want the probability of those, all those 10 events. 499 00:59:21.599 --> 00:59:25.829 1st, 5 events under a quarter of the next 5 events greater than a half. 500 00:59:25.829 --> 00:59:31.679 The intuitive thing just works here, there's nothing tricky or complicated about this. 501 00:59:31.679 --> 00:59:38.429 And so what we get is a probability. 502 00:59:38.429 --> 00:59:42.360 Of a number being less than a quarter is 1 quarter. 503 00:59:42.360 --> 00:59:45.360 Probably being greater than a half is 1 half. 504 00:59:45.360 --> 00:59:54.059 And this will be here so, 5 times you got less than a quarter times. 5 times we got greater than a half. So, 4 to the 5th time to have to the. 505 00:59:54.059 --> 00:59:57.690 Nothing tricky there at all. Yeah. 506 00:59:57.690 --> 01:00:09.179 Okay, so Here's some terminology that I mentioned informally now, let's nail it down. We've got different types of laws. 507 01:00:11.429 --> 01:00:15.659 I've trial is we toss a coin. 508 01:00:15.659 --> 01:00:21.690 Once, and but maybe the coin is not a fair coin. 509 01:00:23.699 --> 01:00:29.849 So that it's quite simply and then we've got the probably that came up heads some P or something. 510 01:00:31.230 --> 01:00:34.500 And we can talk about a sequence of. 511 01:00:36.179 --> 01:00:47.039 So, this is really trial, and then we can get things that we did a sequence of renewal trials, like 10 in a row or something and we get something called button binomial probabilities. 512 01:00:47.039 --> 01:00:51.960 So, what's happening here is so each separate. 513 01:00:51.960 --> 01:01:02.250 Coin toss, we just called and then the sequence of them are getting probabilities that are called binomial probabilities. We saw that before to sing it again. More formally. Now. 514 01:01:02.250 --> 01:01:05.429 So, maybe we tossed the coin 3 times. 515 01:01:05.429 --> 01:01:17.429 And a sequence of 3 probably 3, but normally trials, and now we're interested in these binomial things such as the probability of 3 heads. 516 01:01:17.429 --> 01:01:21.300 Attended or the probably that we got specifically had had tail. 517 01:01:21.300 --> 01:01:31.050 All the 1 of the previous examples of head and ahead. And a Taylor head is probably P tail is probably 1 minus 3. so, this is the probability that we got specifically. 518 01:01:31.050 --> 01:01:37.170 Head and head and tail. This is probability. We got specifically head and tail it had and so on. 519 01:01:37.170 --> 01:01:42.539 So, now we can start adding up there's 3 ways that we could get 1 head and. 520 01:01:42.539 --> 01:01:46.289 Who tails Dale tail head tail head tail. 521 01:01:46.289 --> 01:01:50.250 And tale tale, so it's time 3 here. 522 01:01:50.250 --> 01:01:56.190 And this is a probability that we got 1 head and 2 tails that we don't care where the head came. 523 01:01:56.190 --> 01:02:01.199 And this 3 here, this is this thing. 524 01:02:01.199 --> 01:02:06.269 Probability that we got 2 heads, and we don't care which 2 of the 3 were heads. 525 01:02:06.269 --> 01:02:11.070 We got all 3, so. 526 01:02:11.070 --> 01:02:15.030 And this here, this is this formula right here. The probability. 527 01:02:15.030 --> 01:02:21.420 That we talked to coin in times and we got exactly K heads, but we don't care which. 528 01:02:21.420 --> 01:02:24.630 Of the, and where the K heads. 529 01:02:24.630 --> 01:02:32.610 So that is this thing here and choose K that we saw in factorial divided by K back, tomorrow and minus. K. in fact. 530 01:02:34.019 --> 01:02:39.539 Okay, binomial probability law, write it down a new cheat seat memorize it whatever. 531 01:02:41.280 --> 01:02:47.280 Oops, come on. I'm here. 532 01:02:51.329 --> 01:02:55.230 Okay. 533 01:02:56.250 --> 01:03:06.809 Good everyone. Okay. So and this is this number of ways to choose K items from it. 534 01:03:09.269 --> 01:03:17.340 And we can work things out and so on. So, and there's ways to compute it. I'm going. 535 01:03:17.340 --> 01:03:23.730 Light on that work out ways to compute it if you want, but. 536 01:03:23.730 --> 01:03:27.960 On 1, interesting thing is that if you sum the things up. 537 01:03:27.960 --> 01:03:31.800 Let me show you. 538 01:03:35.130 --> 01:03:45.090 If we sum up 2 to the end. 539 01:03:46.559 --> 01:03:56.099 An equal 3. 540 01:03:59.130 --> 01:04:05.190 It was 1, and some people say it was too. 541 01:04:05.190 --> 01:04:08.190 So. 542 01:04:08.190 --> 01:04:12.900 How you would prove it is. 543 01:04:14.519 --> 01:04:22.230 Well, if we take that thing, well, from the binomial theorem, actually. 544 01:04:22.230 --> 01:04:35.340 So, we have a plus B to the end. 545 01:04:35.340 --> 01:04:41.400 Equals the sum of all the caps at choose K. 546 01:04:43.739 --> 01:04:47.489 Okay. 547 01:04:49.710 --> 01:04:58.199 It goes 1, so 2 to the, and it calls the sum of all the interest K. 548 01:04:58.199 --> 01:05:01.530 That was 1 to the end. Okay. 549 01:05:03.329 --> 01:05:08.909 You proof right there if you accept binomial therapy. Oh, okay. 550 01:05:08.909 --> 01:05:19.469 Now, how you compute this in practice, I'm going to skip there are ways to do that and so on. 551 01:05:21.719 --> 01:05:36.030 Now, 239 here, I hit this example before a little I'm going to come back to this. This is you have some sort of you're providing. 552 01:05:36.030 --> 01:05:39.119 You know, your cell phone company, perhaps. 553 01:05:39.119 --> 01:05:44.760 Your providing is a question of how many channels do you have to build. 554 01:05:44.760 --> 01:05:51.570 So that this particular example, we have 8 customers. 555 01:05:51.570 --> 01:06:00.300 We assume that they're independent of each other and each customer has at any given. 2nd. 556 01:06:00.300 --> 01:06:04.380 As a 4th chance that that customer wants to use the channel. 557 01:06:05.670 --> 01:06:09.480 Now, let's suppose that we have only 6 channels. 558 01:06:09.480 --> 01:06:15.239 For the 8 customers, because the company's trying to save money. 559 01:06:16.469 --> 01:06:20.969 So, then the, the company wants to. 560 01:06:20.969 --> 01:06:32.909 Calculate, what's the probability that they're going to happen? Unsatisfied customer, which would mean that 7 or 8 customers want to talk at the same time? 561 01:06:32.909 --> 01:06:36.719 And that's going to be a problem because this only 6 channels. 562 01:06:37.949 --> 01:06:45.090 So, we can use our column so this, so what we want to do, we want to find the probability that. 563 01:06:45.090 --> 01:06:51.300 There were 7 customers, plus the probability that 8 customers exactly. Wanted to talk. 564 01:06:51.300 --> 01:06:55.469 Again, each customer independently is the 4th chance of wanting to talk. 565 01:06:55.469 --> 01:06:59.849 Okay, so the probability of 7 customers exactly. 566 01:06:59.849 --> 01:07:08.429 Well, 8, 2, 7 could be any the 7 customer will the silent won't be any 1 of the talkie 1 to be any 7 of the 8. 567 01:07:08.429 --> 01:07:14.369 So that's a, to 7 ways we could have 7 of the customers talking. 568 01:07:14.369 --> 01:07:18.000 Any 1 customer 4th chance. 569 01:07:18.000 --> 01:07:32.909 4th to the 7th probability that there are 7 talking customers times 2 thirds are probably with 1 silent customer. So this thing here is the probability that some 7 of the 8 customers want to talk. We don't care which 7. 570 01:07:32.909 --> 01:07:38.130 Plus the probability that all 8 on a talk, well, that's just 4th to the. 571 01:07:38.130 --> 01:07:44.340 So there's a point 2% chance at exactly. 7 want to talk plus. 572 01:07:44.340 --> 01:07:50.010 Find a 1% chance that all 8 want to talk and this nets out here. 573 01:07:50.010 --> 01:07:56.219 So, roughly 1 time in 400, we're going to have an unhappy customer. 574 01:07:57.449 --> 01:08:01.710 Now, you have to decide is, is that number acceptable. 575 01:08:01.710 --> 01:08:09.659 So, Eva. 576 01:08:11.309 --> 01:08:18.930 Here's the related example to that. What's happening here with 2 for example 240. 577 01:08:18.930 --> 01:08:24.119 Is that we have a noisy channel. 578 01:08:25.350 --> 01:08:29.279 Maybe because 7 customers are trying to talk at the same time or something. 579 01:08:29.279 --> 01:08:36.989 And we're going into a really simple error correction method and the really simple method is, they're going to transmit each. 580 01:08:36.989 --> 01:08:41.069 3 times thrice grace means 3 times. 581 01:08:41.069 --> 01:08:47.729 Like, twice means 2 times. Okay. There's no twice for 4 times. However, so we're going to transmit each bit. 582 01:08:47.729 --> 01:08:51.840 3 times, so it's going to take 3 times the channel capacity. 583 01:08:51.840 --> 01:08:55.289 But we're assuming perhaps channel capacity is cheap. 584 01:08:55.289 --> 01:09:03.779 And this doesn't take much computation so we transplant each bit 3 times and the receiver does the majority vote. 585 01:09:03.779 --> 01:09:09.840 And, you know, there's an odd number of bits received, so there's never a tie. 586 01:09:09.840 --> 01:09:17.340 And whichever bet goods receive twice will guess so that's the 1 that was transmitted. 587 01:09:18.479 --> 01:09:25.859 But maybe that's going to go wrong. It's going to go wrong if 2 or 3 bits were where got scrambled in transmission. So. 588 01:09:25.859 --> 01:09:36.390 What's the probability and this thing throws in actual numbers it assumes that there's no point 1% chance that a bit. 589 01:09:36.390 --> 01:09:45.630 Is transmitted badly and we're also assuming that doesn't matter if the transmitted bit was a 0T or 1 they each have an equal probability of going bad. 590 01:09:45.630 --> 01:09:49.170 Why not be true in the real world actually, but. 591 01:09:49.170 --> 01:09:52.590 We don't have any contradictory evidence. Well, so much true. 592 01:09:53.789 --> 01:09:57.359 So, what's the probability that a bit got received badly? 593 01:09:57.359 --> 01:10:02.460 So, it was translated thrice and so if 2 of the 3 times. 594 01:10:02.460 --> 01:10:07.920 We're went bad we don't care which 2 so there's 3 ways that could occur. 595 01:10:07.920 --> 01:10:15.960 O1 squared that those 2 ones went bad times point 9 9 9 that all 3 were good. So this here. 596 01:10:15.960 --> 01:10:20.729 Is the probability the 2 of the 3 transmissions were bad? Exactly. 2 of the 3. 597 01:10:20.729 --> 01:10:24.479 This is the probability that all 3 went bad. 598 01:10:24.479 --> 01:10:27.600 And it's quite just point of all 1 cube. 599 01:10:28.829 --> 01:10:32.850 And you add them up and you get roughly well, it turns out that. 600 01:10:32.850 --> 01:10:37.109 The 1st term totally dominates the 2nd term and it's. 601 01:10:37.109 --> 01:10:42.810 3 times in a 1M that will receive the bit bad. So. 602 01:10:42.810 --> 01:10:51.029 We started out with 1 time and a 1000 that the bit would we received wrong and we approved it to 3 times in a 1000000. 603 01:10:51.029 --> 01:10:55.859 We improved it by a factor of 300. that's a good error correction scheme. 604 01:10:55.859 --> 01:10:58.890 You know, so. 605 01:11:00.449 --> 01:11:04.170 This is this error correction coding and chose using common. 606 01:11:04.170 --> 01:11:07.859 Now, real world. 607 01:11:07.859 --> 01:11:13.409 Correction schemes are much more sophisticated than this. 608 01:11:13.409 --> 01:11:16.770 As buzzwords, like read Solomon and so on. 609 01:11:16.770 --> 01:11:26.909 And, but they're also more complicated, but this is a very simple thing because it's a profitability course, not a communication scores. 610 01:11:29.430 --> 01:11:38.010 Okay, let's look here probability of this and so okay, our error correction. 611 01:11:38.010 --> 01:11:43.529 We saw a multi nomia we're working on way up to geometric and so on. 612 01:11:43.529 --> 01:11:49.020 The multi nomia we saw this before, and just to refresh you. 613 01:11:49.020 --> 01:11:53.609 We have impossible output. We tossed to die. 614 01:11:53.609 --> 01:11:57.510 And we tossed 1 die, it's got 6 paces. 615 01:11:57.510 --> 01:12:00.510 6 possible outputs M is 1. 616 01:12:00.510 --> 01:12:05.100 And we want to know and we, but we toss the dye. 617 01:12:05.100 --> 01:12:15.600 A number of times we tested it 10 times and we want the probability that we saw some. 618 01:12:15.600 --> 01:12:28.109 Combination of we saw 1, 1 and 2 twos and 3 three's and 4 fours and no zeros are no Pfizer sexes or something. And this is the formula right there. And I showed you how you could calculate it. 619 01:12:30.119 --> 01:12:36.060 Okay, so binomial system is too. I did the dark example, so. 620 01:12:36.060 --> 01:12:39.359 Picking phone numbers at Rand. 621 01:12:39.359 --> 01:12:43.710 This is cool. I used a plane crash example. 622 01:12:45.090 --> 01:12:48.630 I don't know phone books if you have seen when I ran, but. 623 01:12:48.630 --> 01:12:54.119 We picked we picked 10 digits of 0T to 9. we pick each. We do 10 picks of that. 624 01:12:54.119 --> 01:12:58.500 What's the probability that we got? Each digit? Exactly once. 625 01:12:58.500 --> 01:13:03.750 But we don't care what order this is a multi thing of. 626 01:13:03.750 --> 01:13:07.920 Each of the 10 possible outcomes occurred once. So this is. 627 01:13:07.920 --> 01:13:12.210 The last thing as point 1 to the 10, because. 628 01:13:12.210 --> 01:13:16.770 It's digital point 1, chance of occurring and this is. 629 01:13:17.819 --> 01:13:25.619 1 chance in 300, so if you pick a digit 10 times, almost always, you're going to get at least duplication and so on. 630 01:13:25.619 --> 01:13:32.640 Generalization of the generalization of the birthday problem. 631 01:13:32.640 --> 01:13:39.659 Okay, so next up is geometric probability, but it's a quarter after now. So. 632 01:13:39.659 --> 01:13:47.279 I'm going to do this on Monday, and so, Monday we'll continue on geometric probability law. 633 01:13:47.279 --> 01:13:56.609 So, just to review what I showed you today was several examples on base law, because it's very important. 634 01:13:56.609 --> 01:14:05.250 And so it's good to understand that examples with communications theory and so on, and then hitting on things like. 635 01:14:07.020 --> 01:14:13.109 Binomial multi nomial is 1 toss of the coin. Binomial is. 636 01:14:13.109 --> 01:14:18.239 K. of a toss and end times. You got K heads you don't care what order. 637 01:14:18.239 --> 01:14:23.670 And multi, normally your tossing the die, and you got some distribution of. 638 01:14:23.670 --> 01:14:34.590 Face is coming up, but you don't care what order geometric. I'll just tease you a little here. The experiment is that you talk to the coin until you get ahead. 639 01:14:34.590 --> 01:14:41.789 And you'll keep going until you get the head, no matter how long it takes. So that's a geometric probability. Law. 640 01:14:41.789 --> 01:14:49.619 50% chance you get it on the 1st toss 25% chance. The 1st had occurs in the 2nd toss. 641 01:14:49.619 --> 01:14:53.880 0T.1% chance the 1st head occurs on the 10th and so on. 642 01:14:56.819 --> 01:15:00.810 But reasonable point to stop now and so. 643 01:15:01.979 --> 01:15:06.000 If there's any questions I'll stay around for a few minutes, if not. 644 01:15:06.000 --> 01:15:10.710 Have a good weekend, go up, get some exercise and. 645 01:15:10.710 --> 01:15:15.270 I might be skiing this weekend again, maybe cross country skiing so. 646 01:15:17.279 --> 01:15:21.899 And I'll upload everything. Well, isn't it. 647 01:15:21.899 --> 01:15:28.050 Silence. 648 01:15:28.050 --> 01:15:32.069 There's questions that. 649 01:15:36.750 --> 01:15:39.750 So, using the iPad. 650 01:15:39.750 --> 01:15:43.170 Is actually working out tolerably apart? 651 01:15:43.170 --> 01:16:03.090 Silence 652 01:16:03.324 --> 01:16:04.345 really quick. 653 01:16:04.619 --> 01:16:08.130 Topics that will be on the exam. 654 01:16:08.130 --> 01:16:13.260 Well, it'll be everything up until that I've covered. 655 01:16:15.270 --> 01:16:20.039 Up until like the class to classes before the exam. 656 01:16:20.039 --> 01:16:23.789 And the way we get exam questions. 657 01:16:23.789 --> 01:16:28.979 Is by going through the blog and so on. So. 658 01:16:28.979 --> 01:16:32.520 And recycling questions from the. 659 01:16:34.260 --> 01:16:39.659 homeworks, of course, no. 660 01:16:41.310 --> 01:16:44.489 Okay, so. 661 01:16:50.159 --> 01:16:54.930 Yes, I did it. 662 01:16:54.930 --> 01:16:58.380 You're you're breaking up, could you say that again? 663 01:16:58.380 --> 01:17:03.270 So, I send you a message in your chat. I'd like to look at it. 664 01:17:03.270 --> 01:17:08.010 About the exam? No about the homework. 665 01:17:08.010 --> 01:17:14.550 Because I didn't know when homework 3 was due. So, like, where is it? I don't say it in the chat. 666 01:17:15.175 --> 01:17:29.755 Oh, you mean the Webex meeting thing oh, okay. Yeah, I wasn't reading it during the class. All homeworks. Well, nothing was due Monday since that was a holiday. If you're in a place, which is having. 667 01:17:30.029 --> 01:17:36.210 Interesting weather like Texas where you about your electricity and we'll. 668 01:17:36.210 --> 01:17:41.850 You know, we'll take the homeworks after you get several days after you get electricity back. 669 01:17:41.850 --> 01:17:46.079 Was that your question? Oh. 670 01:17:46.079 --> 01:17:50.430 We did have homework to Monday. Well, I postponed it. 671 01:17:50.430 --> 01:17:56.460 I think so. It was home at 3 and it's. 672 01:17:56.460 --> 01:18:03.090 Is 2 says February 15 they were good. They were going to accept of light so I think so. 673 01:18:03.090 --> 01:18:06.119 Because that was because the 15th was a holiday. 674 01:18:06.119 --> 01:18:09.600 It be like, I cannot submit anything. 675 01:18:11.010 --> 01:18:15.510 Email me to remind me and we'll, we'll make it we'll open it up again. So. 676 01:18:15.510 --> 01:18:22.199 Or bring it up. Let me just see Webex teams there. 677 01:18:23.819 --> 01:18:29.220 All right it's a 2nd. 678 01:19:19.949 --> 01:19:24.090 Okay, I'll I'll have it extended though, so you'll be okay on that. 679 01:19:24.090 --> 01:19:29.520 Okay, thank you email me to remind. 680 01:19:29.520 --> 01:19:35.609 Could you did I say when the exam would be my yeah, I put it up on the. 681 01:19:38.399 --> 01:19:42.479 In a week and a half or 2 weeks. So. 682 01:19:42.479 --> 01:19:46.140 Archer. 683 01:19:46.140 --> 01:19:50.039 And we're gonna give it twice because of people in other time zones. So. 684 01:19:52.079 --> 01:19:56.729 Okay, let's. 685 01:20:01.260 --> 01:20:06.630 Silence. 686 01:20:06.630 --> 01:20:13.560 Silence. 687 01:20:16.859 --> 01:20:22.409 Okay, if there's nothing else, then see, you. 688 01:20:22.409 --> 01:20:29.819 1 day no 1 else.