WEBVTT 1 00:01:52.890 --> 00:01:55.950 Um, we don't hear anything from you yet. 2 00:02:47.669 --> 00:02:51.060 Is this any better? Does this work now? 3 00:02:51.060 --> 00:02:54.300 Yeah, so we can hear you now. Okay. Thank you. 4 00:02:54.300 --> 00:03:03.060 And if the volume goes crazy, tell me if you're curious what I'm doing, cause I've got 2 machines and profit for me. 5 00:03:03.060 --> 00:03:08.759 And I got the microphone and 1 machine turned on and I'm sharing the video from the 2nd 1. 6 00:03:08.759 --> 00:03:14.189 Okay, and I have a chat window open so that I can, um. 7 00:03:14.189 --> 00:03:24.180 See comments from you. Okay, so what is happening now today is basically. 8 00:03:27.780 --> 00:03:30.960 We're going into chapter 2 of the. 9 00:03:32.460 --> 00:03:36.659 Not the, all of it, of course, is a very big chapter. Chapter. 1 was in the production. 10 00:03:36.659 --> 00:03:41.189 And chapter 2 is getting more solid, but now, 1st, um. 11 00:03:41.189 --> 00:03:44.490 I have if you look on. 12 00:03:44.490 --> 00:03:47.909 The blog here in the top menu, there's a files button. 13 00:03:47.909 --> 00:03:51.719 And this has files for the course such as. 14 00:03:51.719 --> 00:04:00.750 It will have the when the videos saved, we're doing it virtually the video. If it works, it doesn't always work. 15 00:04:00.750 --> 00:04:06.030 The handwritten notes from last time, and so on. 16 00:04:06.030 --> 00:04:09.629 Okay. 17 00:04:09.629 --> 00:04:19.829 No, so rather than hand writing too many things, I'm going to largely. 18 00:04:22.410 --> 00:04:28.468 Finish off from just some chapter 1 stuff, move into chapter 2. 19 00:04:28.468 --> 00:04:42.238 That there is a point cost output you can play with if you wish that, as an example of how you converge with your cost, the coin many times. 20 00:04:42.238 --> 00:04:46.649 If we toss 5 times and. 21 00:04:46.649 --> 00:04:50.939 I got this time for you to to talks again. 22 00:04:53.668 --> 00:04:58.139 And so on, it's different amounts every time. 23 00:04:58.139 --> 00:05:02.999 And and you can run it on your own, but basically. 24 00:05:02.999 --> 00:05:14.999 If you have a small number of as it could vary from the meeting by some amount, but if you get more and more costs, it will tend to cluster. This is like the law of large numbers. 25 00:05:14.999 --> 00:05:29.488 Examples okay, so what I want to do there is what I said Monday is I gave you some general ideas of application to useful. 26 00:05:29.488 --> 00:05:34.528 What somebody else is muted. 27 00:05:34.528 --> 00:05:37.978 Um. 28 00:05:49.379 --> 00:05:52.918 However, okay. 29 00:05:55.408 --> 00:05:59.158 So, if you cannot hear me type something in the chat window. 30 00:05:59.158 --> 00:06:05.098 Again, because I don't get good audio feedback. Audio feedback will start a little here. 31 00:06:05.098 --> 00:06:15.598 Okay, so what I want to do. Hey, thank you. Very good. So, what I want to do now is get some more specific examples. We're probably already is useful. 32 00:06:15.598 --> 00:06:19.559 1 of them is an unreliable channel and this is on page um. 33 00:06:19.559 --> 00:06:23.009 12 of Garcia the textbook. 34 00:06:23.009 --> 00:06:26.728 So, we're trying to spending a bit. It's over a noisy channel. 35 00:06:26.728 --> 00:06:38.158 And the problem is that every so often the bit arrived wrong, it gets mangled and let's say it's all point 1% of the time. So, let's say at 1 time in a 1000. 36 00:06:38.158 --> 00:06:49.228 The bit transmits raw is received wrong and we'll make it symmetric this time. Monday. I talked to him about isometric cases. Keep it simple symmetric. 37 00:06:49.228 --> 00:06:56.158 So, what do you transmit is there or transmit a 1? Then? 1 time in a 1000 it gets flipped. 38 00:06:57.509 --> 00:07:01.019 And that's not good, but maybe it's financial information. 39 00:07:01.019 --> 00:07:06.689 So this very sophisticated ways to do error correction, read Solomon codes and so on. 40 00:07:06.689 --> 00:07:15.119 But Here's a really simple 1. I'll just transmit the bit 3 times. That's thrice. And then. 41 00:07:15.119 --> 00:07:26.158 Do a majority vote, so if I, if it's a 0, I transcript 03 times, if it's received correctly, 3 times and receive versus 0. 42 00:07:26.158 --> 00:07:32.399 There's 1 transmission error, so 00 ones received or 1 receiver. 43 00:07:32.399 --> 00:07:39.149 Says it's 0, it says 2 transmission errors or 3 transmission areas, then the receiver gets it wrong. 44 00:07:40.678 --> 00:07:55.644 So, we want to so, it's, it's a random thing there. Every time we transmit a bit, it's a different random experiment, and the outcome is, what is the received bid? So we can throw some math at the problem now and we can compute. 45 00:07:55.644 --> 00:08:00.144 What's the probability that this channel is now wrong? Initially it was 1 time and a 1000. 46 00:08:00.689 --> 00:08:07.439 Well, it ease the probability of an error, so 3 bits arrive correct? It's a 1 minus code. 47 00:08:07.439 --> 00:08:11.428 If there are 1 error, are there's 3 places that could occur. 48 00:08:11.428 --> 00:08:16.319 Um, if it's the 1st, transmission is wrong or the 2nd or the 3rd. 49 00:08:16.319 --> 00:08:20.879 And we can, I mean, this may be reasonably obvious. 50 00:08:20.879 --> 00:08:24.838 That how you could, um, what this is so that. 51 00:08:24.838 --> 00:08:39.239 What the error is so, and we can capture the probability. 2 of the errors are the problem of all 3 errors being wrong. So the corrected is received is 3 bits or 2 bits arrive. Correct we add those 2 numbers point 997 plus point. 52 00:08:39.239 --> 00:08:42.269 409, 94, and we get. 53 00:08:42.269 --> 00:08:47.038 Point 5, nines, 6 guides and the 3 9. 54 00:08:47.038 --> 00:08:50.908 And now, this is basically point 5, 9, 7. 55 00:08:50.908 --> 00:08:58.259 And so this method reduced the transmission, or the probability of an aeroplane with a backpack or 1000. 56 00:08:58.259 --> 00:09:03.688 Which is good what the cost is, we have to transmit 3 times as much stuff. 57 00:09:03.688 --> 00:09:09.688 And we have some more logic, but this is an example. 58 00:09:09.688 --> 00:09:13.649 We have a random experiment with real practical value and we use. 59 00:09:13.649 --> 00:09:21.239 This random experiment, and then we decide something and error correcting transmission and have a new random experiment. 60 00:09:21.239 --> 00:09:28.019 The new random experiment is we transmit a bit 3 times that the outcome is what. 61 00:09:28.019 --> 00:09:35.009 The computer received, Here's another example. Um. 62 00:09:35.009 --> 00:09:38.519 Tax compression. 63 00:09:38.519 --> 00:09:41.908 Now, transmission lines are very fast. 64 00:09:41.908 --> 00:09:51.119 So, you know, you can upload a couple, 100 megabits the 2nd, if you buy the expensive plans, say with, uh. 65 00:09:51.119 --> 00:09:54.839 Specter or Verizon us or something. 66 00:09:54.839 --> 00:09:58.318 But there was a time when transmission lines were very slow. 67 00:09:58.318 --> 00:10:02.129 And the 1st, for example. 68 00:10:02.129 --> 00:10:08.038 Trans Atlantic cable, which is the 850 s from Ireland to land. 69 00:10:08.038 --> 00:10:13.109 Its transmission speed was under 1 bit for a 2nd, I believe. 70 00:10:13.109 --> 00:10:24.719 1st telegraph lines were a little earlier, I believe, the 840 s or something and it was something from Washington D. C. Baltimore or something like that. It was Samuel 10 degrees more. 71 00:10:24.719 --> 00:10:29.278 His contribution actually was the Morse code. 72 00:10:29.278 --> 00:10:32.908 So, again, transmission was relatively slow. 73 00:10:34.619 --> 00:10:39.359 And so he wanted a way to encode letters and tidbits. 74 00:10:39.359 --> 00:10:43.349 In a way that minimized transmission time. So. 75 00:10:43.349 --> 00:10:47.788 The pipe fits for ladder 8 5. 0. 76 00:10:47.788 --> 00:10:53.129 Um, but the problem is, or the property of the English language is, of course, to some letters, like E. 77 00:10:53.129 --> 00:10:57.479 Are more common than other letters such as cute. So what. 78 00:10:57.479 --> 00:11:01.078 More is used a variable length encoding. 79 00:11:01.078 --> 00:11:09.239 Where the common letters had a shorter, uh, fewer bits been used in the railroad letters. 80 00:11:09.239 --> 00:11:15.808 And he was 1 doc, because the doc was the quickest way to transmit and a queue was dash dash. 81 00:11:15.808 --> 00:11:19.528 Dot dash the top. 82 00:11:20.548 --> 00:11:31.198 And thereby what Morris did is, he shortened the average time it would take to translate it. Cool. So, the probability here, the random experiment in this case. 83 00:11:31.198 --> 00:11:43.168 Was a random messages that the customer would want transmitted and the outcome would be the number would be the time it took the telegraph operator to transmit the message. 84 00:11:44.519 --> 00:11:48.989 I have a couple of besides here about probability and so on. 85 00:11:48.989 --> 00:11:52.499 A really good barcode or is faster than texting. 86 00:11:52.499 --> 00:12:00.778 And you can compress English easily down to 2 bits per ladder and with difficulty you can get it down to 1 bit provider. 87 00:12:00.778 --> 00:12:07.139 With a sophisticated compression technique, it's using structured in this language such as. 88 00:12:07.139 --> 00:12:13.798 I mentioned the Monday that queue is almost always followed by you if you have key followed by age the next. 89 00:12:13.798 --> 00:12:18.479 It's going to be a powerful probably in a year or not. 90 00:12:18.479 --> 00:12:21.658 Most of the time could be other things in or are you, or. 91 00:12:21.658 --> 00:12:29.308 Maybe a, why it could be died, but not by clink and you can use this structure and this is how analysis works. 92 00:12:31.708 --> 00:12:36.749 Here's another example, random experiment and this is called reliable system design. 93 00:12:36.749 --> 00:12:41.489 And save a nuclear power plant. 94 00:12:41.489 --> 00:12:45.089 And they're really P*** people off if you have a failure. 95 00:12:45.089 --> 00:12:59.489 And so you could do I showed you a simple 3 for crossing a coin on Monday you can have a more complicated tree for nuclear power plant failing. So, maybe there's a Waterloo plan for 50 years old. 96 00:12:59.489 --> 00:13:02.938 Our pipes that are buried in concrete. 97 00:13:02.938 --> 00:13:06.208 They rushed through after 50 years, start leaking. 98 00:13:06.208 --> 00:13:11.369 And water leaks, or an operator is a sleep. 99 00:13:11.369 --> 00:13:15.808 And usually large number of industrial accidents happened in the middle of the night. 100 00:13:15.808 --> 00:13:19.438 And some backup fails, or it was turned off. 101 00:13:19.438 --> 00:13:22.438 She goes through what happened in Chernobyl, then. 102 00:13:22.438 --> 00:13:29.038 Backups some things were turned off for a test. There was a stress test on the system. 103 00:13:29.038 --> 00:13:32.458 And the system tail distress catch on so. 104 00:13:32.458 --> 00:13:35.999 And you can do it, then you can count the probability of a failure. 105 00:13:35.999 --> 00:13:44.188 That you can, you know, try to get probabilities of these areas things. And if you have redundancy and backups. 106 00:13:44.188 --> 00:13:50.249 And that helps if you've got some quite short thing, it may make it worse. And so. 107 00:13:50.249 --> 00:13:58.109 And people do these, these countries, they're very good precise. I mean, people acknowledge that, but they're better than nothing. 108 00:13:59.399 --> 00:14:06.899 If you have something else, an automobile so if I have 2 cars in my garage. 109 00:14:06.899 --> 00:14:10.349 And I want to calculate say the probability that. 110 00:14:10.349 --> 00:14:14.428 You know, I could get to our API for class then. 111 00:14:14.428 --> 00:14:18.509 You see, if 1 car fails, then the 2nd car. 112 00:14:18.509 --> 00:14:21.839 I might drive the 2nd car unless it's also failed. So it's like. 113 00:14:21.839 --> 00:14:25.979 Transmitting the bit 3 times I've got 2 choices here. 114 00:14:25.979 --> 00:14:30.958 It's not a majority both, but all I need is 1 of my 2 cars to succeed. 115 00:14:30.958 --> 00:14:36.058 To be working to get to our so, yeah, I can get their reliable system to sign. 116 00:14:37.948 --> 00:14:42.208 Or you look in a classroom at RPI there's a lot of lights in the ceiling. 117 00:14:42.208 --> 00:14:45.359 But suppose it's 20 lights up there in the ceiling. 118 00:14:45.359 --> 00:14:50.399 1 or 2, or 3 of the lights fail we still have enough light to run the class. 119 00:14:50.399 --> 00:14:54.058 If 10 of the lights fail, then. 120 00:14:54.058 --> 00:15:00.688 Uh, okay. 121 00:15:02.548 --> 00:15:06.688 I've got a question here from who. 122 00:15:08.369 --> 00:15:11.908 You can hear it's okay. I can't really learn. 123 00:15:11.908 --> 00:15:21.509 How to do questions when I just read off? Oh, okay. I'll do some questions in more detail. Do some questions in more detail later actually. 124 00:15:24.028 --> 00:15:29.908 If you like, um, let's try a question that. 125 00:15:29.908 --> 00:15:33.928 Let's see, what would you like it to be? Um. 126 00:15:38.188 --> 00:15:41.219 Let's do a failure analysis let's say. 127 00:15:41.219 --> 00:15:48.658 Um, let's say it, let me say it in the classroom thing. Let's do 1. 128 00:15:48.658 --> 00:15:53.849 Give me a 2nd, here. 129 00:17:26.818 --> 00:17:31.439 Okay, we have some issues here with shared content. 130 00:17:39.118 --> 00:17:44.608 Okay, my message 1 is not working. Let me try. 131 00:17:44.608 --> 00:17:49.769 A message to here, give me a 2nd. 132 00:17:56.243 --> 00:18:07.074 Okay. 133 00:18:08.273 --> 00:18:08.784 See, 134 00:18:08.784 --> 00:18:09.773 if this, 135 00:18:34.763 --> 00:18:37.763 I'm having a small issue here, 136 00:18:37.763 --> 00:18:40.044 trying to share a window. 137 00:21:11.368 --> 00:21:15.989 This is. 138 00:21:24.148 --> 00:21:34.259 Not that Charlie. Okay. 139 00:21:48.868 --> 00:21:56.068 No, the problem is that the screen sharing at the moment is not working very well. 140 00:21:56.068 --> 00:22:03.118 So that it's in fact. 141 00:22:17.939 --> 00:22:29.608 Okay. 142 00:22:41.878 --> 00:22:45.269 Okay, okay. 143 00:22:46.949 --> 00:22:50.159 Something is actually happening. 144 00:22:51.239 --> 00:23:00.659 So so, okay, let's do an example of reliability then. 145 00:23:19.648 --> 00:23:33.598 Okay. 146 00:23:36.298 --> 00:23:46.378 Okay, um. 147 00:23:56.038 --> 00:24:01.648 Okay, um, let's say. 148 00:24:16.288 --> 00:24:28.378 Okay, so, um, so let's say, um. 149 00:24:34.979 --> 00:24:40.798 Say I know alpha, but then let's say I've got a 2nd car. 150 00:25:08.423 --> 00:25:10.794 Okay, so I need at least 1 car. 151 00:25:25.259 --> 00:25:31.558 Equal, let's say 1 minus so let's suppose. 152 00:25:31.558 --> 00:25:35.128 That's suppose the 1st car has a. 153 00:25:35.128 --> 00:25:42.689 Um, point 01, chance of failure the 2nd car. 154 00:25:44.308 --> 00:25:48.058 As may be a point 02, chance of failure. Um. 155 00:25:48.058 --> 00:25:56.638 I'm on here. 156 00:26:04.199 --> 00:26:12.148 Hello. 157 00:26:14.159 --> 00:26:24.179 Okay. 158 00:26:33.148 --> 00:26:39.479 No, so the car working, probably of a car working there will be 1 minus. 159 00:26:50.608 --> 00:26:53.759 And you can work that out. That's, um. 160 00:27:04.048 --> 00:27:07.858 Okay, so that's an example of doing things like. 161 00:27:07.858 --> 00:27:12.538 Or do it working out a problem with failure. 162 00:27:19.169 --> 00:27:27.358 Okay, so the thing with a. 163 00:27:28.588 --> 00:27:32.098 Um, so that's an example of how you would do a problem like that. 164 00:27:45.358 --> 00:27:49.769 Mm, hmm. 165 00:28:04.648 --> 00:28:11.068 Okay. 166 00:28:35.489 --> 00:28:39.118 I'm giving another example of something like that. 167 00:28:39.118 --> 00:28:48.148 Is so. 168 00:28:48.148 --> 00:28:51.419 Another 1 is. 169 00:29:05.729 --> 00:29:11.128 Give me a 2nd, here in that in my attempt to. 170 00:29:11.128 --> 00:29:14.848 Um, run that and. 171 00:29:15.898 --> 00:29:21.088 Something else got something else got mangled here. 172 00:29:21.088 --> 00:29:34.558 So that I have to. 173 00:30:01.709 --> 00:30:06.509 I'm on here, we are having some issues here with. 174 00:30:06.509 --> 00:30:14.338 Sharing the screen. Well. 175 00:30:14.338 --> 00:30:18.509 Okay, sharing stopped working when I went to share. 176 00:30:18.509 --> 00:30:24.868 To that something else, um, something else like crazy. 177 00:30:26.128 --> 00:30:38.848 Okay. 178 00:30:38.848 --> 00:30:43.499 Any case, so that, um. 179 00:30:48.209 --> 00:30:57.179 So, as an example liable design, now, um, when we get into chapter 2, we get into. 180 00:31:03.179 --> 00:31:08.638 What we get into are some basic concepts of, um, probability theory. 181 00:31:08.638 --> 00:31:18.118 And we get into things such as, um, um, some basic theory things such as set theory. 182 00:31:35.999 --> 00:31:42.868 So, for example, you've got sets of discrete elements, you might have something like, um. 183 00:31:44.128 --> 00:31:51.538 We might have say, tossing a dye as he might have some general sample space. 184 00:31:56.278 --> 00:32:06.598 And then you might have a subset a, which is a 13 going to set B, which was a 1. 185 00:32:08.489 --> 00:32:15.088 56, and then you get a say, you and B, which was. 186 00:32:20.939 --> 00:32:24.898 And so on, she got some basic set theory and. 187 00:32:24.898 --> 00:32:28.769 With probability, this would be a discreet set. 188 00:32:30.509 --> 00:32:40.588 Hello. 189 00:32:49.048 --> 00:32:53.398 Now, you also I was probability you have the continuous sets. 190 00:32:56.278 --> 00:33:03.778 There's continuous sets and for example, suppose the observation is so your experiment. 191 00:33:07.169 --> 00:33:15.179 Is, um, oh, so dark let's say. 192 00:33:21.628 --> 00:33:26.578 Pass it art and then, so what would be some problems? So maybe. 193 00:33:26.578 --> 00:33:29.909 You might have an event or something. 194 00:33:32.729 --> 00:33:40.169 1 would be 1.7 or whatever. 195 00:33:43.648 --> 00:33:47.729 And would be. 196 00:33:51.449 --> 00:33:58.288 Then you go 2.5 so then the question would be, you might have. 197 00:33:58.288 --> 00:34:03.358 1, and that's going to be. 198 00:34:04.618 --> 00:34:09.088 Instead of all X sets that. 199 00:34:15.748 --> 00:34:26.068 Okay, so you would have, um, you could actually upset theory here. Um, 2nd, point in. 200 00:34:28.079 --> 00:34:41.603 I'm in chapter 2 is you're going to get axioms. Axiom is a probability. 201 00:34:42.474 --> 00:34:46.463 Axiom is a general set of. 202 00:34:48.809 --> 00:34:57.088 It's a general set of rules that have abstracted away from from specific things. So if, um. 203 00:34:57.088 --> 00:35:01.528 So, P1 is the probability of something we're going to say. 204 00:35:10.469 --> 00:35:22.768 We're going to have some rules such as, um, it's between 0 and 1 and if we've got 2 destroys events, um. 205 00:35:31.918 --> 00:35:43.378 Probabilities add and so on so. 206 00:35:43.378 --> 00:35:48.989 If we're talking about, say, coin taught, I mean, say a diet dye. 207 00:35:51.778 --> 00:35:58.349 Um, let's say so event 1 is. 208 00:35:58.349 --> 00:36:03.509 Say, 1 and 2 and the probability is 4th. 209 00:36:03.509 --> 00:36:12.208 Let's say even 2 is that you saw 3 or 6. 210 00:36:12.208 --> 00:36:17.489 Is 4th and then. 211 00:36:18.688 --> 00:36:31.498 Event 1, or event 2, the probability is going to be probably 1 plus probability to equals 2 thirds. And this is true because the, um. 212 00:36:33.358 --> 00:36:37.318 Because the 2 events were destroyed, so the probabilities add. 213 00:36:39.088 --> 00:36:46.228 So, we're going to see like this, and we're going to see things like the sample space. 214 00:36:51.989 --> 00:36:56.699 So the sample space would be all possible outcomes. 215 00:36:56.699 --> 00:36:59.699 And for the. 216 00:37:02.039 --> 00:37:12.298 For the die, the sample space is going to be. 217 00:37:12.298 --> 00:37:25.679 Okay. 218 00:37:30.778 --> 00:37:41.219 And for a coin, it'd be heads or tails for the continuous example, which is the uncountable infinity than the sample space. 219 00:37:41.219 --> 00:37:44.909 Would be the interval of real numbers let's say. 220 00:37:44.909 --> 00:37:53.429 Okay. 221 00:38:18.329 --> 00:38:31.110 Okay. 222 00:38:31.110 --> 00:38:39.059 Okay, now it is an important. I'll give you an example example here. 223 00:38:39.059 --> 00:38:42.389 Showing you the thing about experiments. 224 00:38:42.389 --> 00:38:44.335 And observations, 225 00:39:06.985 --> 00:39:07.284 so, 226 00:39:08.215 --> 00:39:09.594 random experiment, 227 00:39:09.655 --> 00:39:10.644 you can have, 228 00:39:10.675 --> 00:39:11.184 um. 229 00:39:11.579 --> 00:39:25.679 Okay, might design. 230 00:39:27.900 --> 00:39:30.989 My design different observations. 231 00:39:33.179 --> 00:39:36.750 So. 232 00:39:45.119 --> 00:39:55.500 I give you an example. Um. 233 00:40:00.239 --> 00:40:09.900 Okay oh, okay. 234 00:40:18.599 --> 00:40:29.789 Now, I will not be using discord. I'll be using the chat window here because, um, it's just 1 more medium to chat window. The chat window does seems okay you're welcome to. 235 00:40:29.789 --> 00:40:35.820 Join things to yeah, you're welcome to talk among yourselves. Obviously. 236 00:40:35.820 --> 00:40:40.739 Okay, so so you might design different observations. 237 00:40:40.739 --> 00:40:45.719 And I give an example, um. 238 00:40:47.369 --> 00:40:54.690 Suppose we have an app. It's a nice example of I picked sodium 26, because sodium 26. 239 00:40:54.690 --> 00:40:58.409 Is has a half life of 3rd. 240 00:40:58.409 --> 00:41:01.860 So, we might say experiment 1. 241 00:41:03.599 --> 00:41:11.849 We might observe, um, the underground number of seconds. 242 00:41:17.550 --> 00:41:24.929 Okay, okay. So here your sample space of outcomes. 243 00:41:24.929 --> 00:41:29.159 Is there a 1 2 3 4. 244 00:41:29.159 --> 00:41:36.389 And so on following up to infinity. 245 00:41:41.610 --> 00:41:45.360 And so, okay. 246 00:41:45.360 --> 00:41:49.079 That by the way is Countable infinite. 247 00:41:52.139 --> 00:41:55.199 And it's also discreet. 248 00:41:58.440 --> 00:42:02.340 We could design a 2nd experiment. Um. 249 00:42:04.019 --> 00:42:14.789 Which could be something like experiment 2 would be, um. 250 00:42:19.710 --> 00:42:31.050 Would be time to that would be, say, X, some accents greater equal to 0. 251 00:42:31.050 --> 00:42:41.820 And this would be a continuous example. So, the underlying thing is the same thing, but it's, um. 252 00:42:44.789 --> 00:42:48.150 But we do, we define different outcome the outcome is what. 253 00:42:48.150 --> 00:42:54.750 Is what we want to measure so now you can start getting, um. 254 00:42:54.750 --> 00:43:02.730 And stop start getting events after the 2nd 1. 255 00:43:02.730 --> 00:43:06.420 Again, you could have an event such FedEx. 256 00:43:06.420 --> 00:43:09.630 Is very difficult 3.5 or something. 257 00:43:09.630 --> 00:43:14.039 And you get probabilities on that. 258 00:43:16.230 --> 00:43:21.059 And I give examples on the blog you could actually, um. 259 00:43:21.059 --> 00:43:27.269 You could actually you could actually, so number that we could say is, um. 260 00:43:33.809 --> 00:43:41.610 And let's say the case, the first second. 261 00:43:44.579 --> 00:43:49.769 So, is there 1, let's say. 262 00:43:50.880 --> 00:43:55.500 And then we can talk about the event that and the, um. 263 00:44:00.090 --> 00:44:06.659 Invented to case, for example, in the 1st, 3 seconds. 264 00:44:08.940 --> 00:44:18.119 Now, how would you do that? Well, this would be in each given. 2nd I mean, I'm anticipating a little here, but in each given 2nd. 265 00:44:19.559 --> 00:44:24.809 The probability that it is, is 1 half. 266 00:44:24.809 --> 00:44:28.349 So it will be 1 half. 267 00:44:29.670 --> 00:44:34.769 That it decayed in the first second if it did not decay in the first second. 268 00:44:34.769 --> 00:44:39.360 That's a 1 half to talk to you in the first second then There'll be in 1 app. 269 00:44:39.360 --> 00:44:47.340 That the case in the 2nd, if it survived the 4th, it will be a probability of 1 quarter that had got. 270 00:44:47.340 --> 00:44:55.650 Through the 1st, 2 seconds times of 1 half that it and the third second. 271 00:44:55.650 --> 00:45:00.360 And we would end up with, um, let's see, here. 272 00:45:01.800 --> 00:45:08.519 We'd end up with a 1 half plus 1 quarter plus 1, 8. 273 00:45:08.519 --> 00:45:19.559 Course, we end up with the 7 X will be cut the probability of something like that. 274 00:45:19.559 --> 00:45:24.989 Okay, I do too. Well, but this is an example that. 275 00:45:24.989 --> 00:45:34.469 We have the random experiment. The random experiment is to look at the sodium outcome and we define various outcomes. The outcome could be. 276 00:45:34.469 --> 00:45:46.019 The discreet thing the number of seconds before does, it became the first second that would be 0 complete seconds. Does it became the 4th that would be 1 complete. 2nd does it became the third second. 277 00:45:46.019 --> 00:45:50.579 And so that would be 2 complete seconds. We've got the different outcomes. 278 00:45:50.579 --> 00:45:54.119 And then we could compute the profitability of an event. 279 00:45:54.119 --> 00:45:57.869 Which is that the case and the 1st, 3 seconds for example. 280 00:46:00.119 --> 00:46:11.579 Um, if we take the same random experiment, and we have different, um, we have different outcomes, which is the exact time until it and we would have different events. 281 00:46:11.579 --> 00:46:14.909 And we calculate probabilities. 282 00:46:16.739 --> 00:46:24.809 But in any case, we got the 2 types of experiment. We've got the outcomes that are discreet and we've got the outcomes that are continuous and. 283 00:46:24.809 --> 00:46:28.829 You could get them from the same experiment, depending on what it is that you decide to measure. 284 00:46:28.829 --> 00:46:34.019 You've got set theory, you've seen some set theory before and so on. 285 00:46:36.389 --> 00:46:41.940 Okay, and the profitability distribute I mentioned there are some examples of, um. 286 00:46:41.940 --> 00:46:45.630 Probabilities go from 0 to 1 and so on. 287 00:46:46.980 --> 00:46:50.789 Okay, let me give you another example of working with some. 288 00:46:50.789 --> 00:46:57.360 Some axioms of probability I have them on the blog. 289 00:46:57.360 --> 00:47:04.920 Let me, um, okay, let me do another 1 here. 290 00:47:09.360 --> 00:47:12.690 Okay. 291 00:47:26.130 --> 00:47:37.559 Let's talk about a deck of cards, so you've got various axioms and probability, which is. 292 00:47:38.969 --> 00:47:43.949 For example, 1 of the axioms is if we've got different events, a, and B. 293 00:47:43.949 --> 00:47:47.429 The probability of a union be close. 294 00:47:51.090 --> 00:47:58.980 Probability of a plus the probability of being minus the probability of a intercept B. 295 00:47:58.980 --> 00:48:02.820 Okay, let me give you let me work out an example for you. 296 00:48:02.820 --> 00:48:14.429 Let's let we're going to work with event queue, which is that we do equate until it gets 52 cards in tech. So, in an event queue. 297 00:48:14.429 --> 00:48:20.130 And the probability that we draw the queue. 298 00:48:20.130 --> 00:48:26.550 Is going to be on 113rd the 52 cards for them are Queens. 299 00:48:26.550 --> 00:48:30.389 We'll have an event and other event, which is. 300 00:48:30.389 --> 00:48:36.420 A hard we're talking about the individual outcomes are which of the 52 cards do we draw. 301 00:48:38.340 --> 00:48:47.309 But, um, let's see, right. 302 00:48:50.994 --> 00:48:58.974 Okay, um, so for the hard, so the individual outcomes are, which of the 52 cards do you draw? 303 00:48:58.974 --> 00:49:08.994 So, the 1st event I'll define as you draw a queen, the 2nd, possible event I'll defined as you draw hard and the probability of a heart is. 304 00:49:10.230 --> 00:49:14.099 1 quarter or 13 of the 52 cards. 305 00:49:15.989 --> 00:49:24.630 Specific in here, the whiteboard is really messing up. Okay. So I want to use 1 of the access to probability the rule of caught up here. 306 00:49:25.710 --> 00:49:28.769 So, let's suppose I want to find a probability. 307 00:49:28.769 --> 00:49:32.309 And I do a queen, or I drew a heart. 308 00:49:32.309 --> 00:49:39.659 Now, in order to do that, I have to find the intersection thing. So what I need. 309 00:49:42.059 --> 00:49:46.050 Is the probability of a queen and a heart. 310 00:49:48.659 --> 00:49:52.590 So someone tell me what would that be? What, what would be the probability. 311 00:49:52.590 --> 00:49:57.119 Of the event, which which a card is both a queen and a heart. 312 00:49:57.119 --> 00:50:01.800 What would that probability? Be something you just type it on the chat window let's say. 313 00:50:03.119 --> 00:50:09.869 Thank you. So now we can use that. 314 00:50:09.869 --> 00:50:16.889 To find this thing up here, remove the question mark. 315 00:50:20.010 --> 00:50:26.130 Say, 100 thirteen's plus 1 quarter minus 152. 316 00:50:26.130 --> 00:50:32.250 And that is going to be 4, plus 13 minus 1. 317 00:50:32.250 --> 00:50:36.570 Which is 650 seconds. 318 00:50:37.829 --> 00:50:42.630 So that's an example of using 1 of the axioms 1 of the rules for profitability. 319 00:50:45.690 --> 00:50:59.519 Okay, now we could do this, um. 320 00:51:01.800 --> 00:51:05.579 We, we could do this, um, in more, um. 321 00:51:06.869 --> 00:51:12.119 In more complicated ways this rule up here is just for. 322 00:51:14.340 --> 00:51:20.670 Um, 2 events, we could take the thing, and we could take a, um. 323 00:51:21.780 --> 00:51:25.530 So you can be again C. 324 00:51:30.119 --> 00:51:35.489 Okay. 325 00:51:37.525 --> 00:51:38.304 Minus 326 00:51:43.644 --> 00:51:57.835 okay, 327 00:51:57.864 --> 00:51:59.034 so. 328 00:52:02.789 --> 00:52:06.750 What would be an example of something like that? 329 00:52:06.750 --> 00:52:17.699 Might be okay. 330 00:52:22.440 --> 00:52:26.280 I'd say we'll work with so you had a, let's say. 331 00:52:27.960 --> 00:52:39.869 That's a queen bee, being it say. 332 00:52:42.030 --> 00:52:51.119 I don't know hey, Jack, a queen or a. 333 00:52:51.119 --> 00:52:54.239 King events see. 334 00:52:54.239 --> 00:53:00.239 Meaning it say, go up or something? 335 00:53:00.239 --> 00:53:04.380 So, now we would need the probability. 336 00:53:04.380 --> 00:53:08.130 Of it being being a queen is 1. 337 00:53:08.130 --> 00:53:12.840 13rd, the probability that was being Jack queen or king. 338 00:53:12.840 --> 00:53:19.260 Is going to be 3 thirteen's the probability of it being a club will be 1 quarter. 339 00:53:20.699 --> 00:53:29.010 The probability of it being a queen and a Jack queen, or king is the 113. 340 00:53:29.010 --> 00:53:35.340 The probability of it being a, or C. 341 00:53:35.340 --> 00:53:45.539 Queen 1, probably with being B or C Jacqueline queen ad club. 342 00:53:45.539 --> 00:53:52.679 352 and the probability of it being all 3. 343 00:53:52.679 --> 00:53:55.920 Um, will be. 344 00:53:55.920 --> 00:54:01.829 152nd, so the probability of the Union. 345 00:54:03.510 --> 00:54:07.170 Will be awesome. 346 00:54:07.170 --> 00:54:15.809 13 plus, plus 1 quarter. 347 00:54:17.190 --> 00:54:26.039 Um, minus 113. 348 00:54:26.039 --> 00:54:31.289 Minus 152 minus. 349 00:54:35.849 --> 00:54:40.469 350 twos plus 10,052. 350 00:54:42.000 --> 00:54:45.719 And if I do everything in 50 seconds. 351 00:54:45.719 --> 00:54:49.590 That will be for us. 352 00:54:49.590 --> 00:55:04.289 12 or 216 plus, um, 13, which is 29 minus 4, which is 25 minus 1, which is 24 minus 3, which is 21. 353 00:55:04.289 --> 00:55:07.920 Plus 1, which is 22. 354 00:55:09.599 --> 00:55:15.539 And you can check me on that. I have a question here. Do I mean, probability. 355 00:55:18.659 --> 00:55:24.510 Um, here at the right it's probably a, and to be. 356 00:55:24.510 --> 00:55:28.320 And see, here, that's what you mean. 357 00:55:28.320 --> 00:55:38.340 Are they pulling from the same card deck? No, I did. This question is that this is we're drawing 1 card. 358 00:55:38.340 --> 00:55:42.059 And we're looking at at different ways so. 359 00:56:01.949 --> 00:56:06.780 We're drawing 1 card here. Okay. Um. 360 00:56:09.840 --> 00:56:15.570 Event B, was that the card that we do it was a Jack or a queen or king? 361 00:56:15.570 --> 00:56:19.860 It can only be 1. yeah. Okay. 362 00:56:19.860 --> 00:56:24.030 No, we're drawing only 1 card, so okay. 363 00:56:24.030 --> 00:56:29.519 I'll work off to a drawing. Let's be okay. 364 00:56:34.019 --> 00:56:40.260 Let me work out some more questions for you now, then 1 questions um. 365 00:56:51.300 --> 00:56:56.820 Okay, so. 366 00:56:58.920 --> 00:57:06.030 So, let's suppose, um, our sample space is, uh, is the interval from 0 to 1. 367 00:57:07.110 --> 00:57:14.340 Okay, and the experiment is to pick a random number from 0 to 1. 368 00:57:14.340 --> 00:57:17.730 And so the probability. 369 00:57:19.349 --> 00:57:23.190 That say a vehicle to be. 370 00:57:23.190 --> 00:57:28.110 Equals and I say. 371 00:57:28.110 --> 00:57:33.750 That, you know, if there are less and less equal to B, let's go 1. 372 00:57:33.750 --> 00:57:37.199 Okay, now. 373 00:57:39.960 --> 00:57:44.340 So event big a, is the interval. 374 00:57:50.699 --> 00:57:56.099 So, the notation here, the square brackets means that the observation. 375 00:57:56.099 --> 00:58:01.739 Was between 2 and 6.2 end points. So the square brackets so event a. 376 00:58:01.739 --> 00:58:05.099 Is that our outcome at some point? 2 2.6. 377 00:58:07.139 --> 00:58:11.730 Event B, is that the outcome was prompt? 378 00:58:11.730 --> 00:58:14.909 Point 4 up to 1. 379 00:58:16.050 --> 00:58:20.309 So, okay, so now here are some questions what's the probability of a. 380 00:58:22.110 --> 00:58:27.329 Anyone okay, so. 381 00:58:31.079 --> 00:58:37.170 Okay, I'm just looking back a question so probably a B and C I supposed to be. 382 00:58:40.949 --> 00:58:45.119 I, I don't understand why do we. 383 00:58:45.119 --> 00:58:48.150 The last part. 384 00:58:50.190 --> 00:58:55.710 Why do we add the last part? Okay, let me go back to answering you a quick. 385 00:58:55.710 --> 00:59:00.210 Okay, I'll let me go back to the previous question. 386 00:59:00.210 --> 00:59:03.900 Before I continue with this then, um. 387 00:59:13.139 --> 00:59:16.829 Okay, well, if we have a here. 388 00:59:18.989 --> 00:59:29.969 And we have the see, and if we want the probability of a B or C, right you got it. 389 00:59:29.969 --> 00:59:35.730 So, we want see. 390 00:59:35.730 --> 00:59:39.030 So that's a here. 391 00:59:41.489 --> 00:59:46.889 B, and C. 392 00:59:46.889 --> 00:59:54.960 And then, okay. 393 00:59:54.960 --> 01:00:00.750 And then we need to do a minus intersect me. That will be here. 394 01:00:00.750 --> 01:00:06.059 And then minus a intersect C That'll be here. 395 01:00:06.059 --> 01:00:10.199 Minus B intersect. C. I'll be here. 396 01:00:10.199 --> 01:00:14.489 And then plus. 397 01:00:14.489 --> 01:00:17.849 Intersect B, intersect see. 398 01:00:17.849 --> 01:00:22.260 So, it was saying in the middle, we took too much out and we ran and pack. So. 399 01:00:22.260 --> 01:00:26.730 And we can, then with the general rule with any of them. 400 01:00:27.869 --> 01:00:33.329 Okay, okay so coming back to this example, here, I'm working out. 401 01:00:33.329 --> 01:00:41.670 That it's a continuous probability example and the sample space is the real numbers from 0 to 1. 402 01:00:41.670 --> 01:00:45.900 And I need I need a rule for probability. 403 01:00:46.434 --> 01:01:00.565 If you know, if physics is a problem, you might determine it, or it might be given to you as part of the problem. So that here so the probability of an interval. So this is a continuous thing appropriate of any specific real number is 0, you can't work with that. 404 01:01:00.565 --> 01:01:07.644 You got to work with the events and the events here, we're going to state that the events are intervals such as the interval from point to 2.6. 405 01:01:08.699 --> 01:01:13.590 Or the interval from point 4 to 1 or something. Okay so that. 406 01:01:15.179 --> 01:01:21.239 And I give a rule for probability, which I said the rule for the probability of an event occurring. 407 01:01:21.239 --> 01:01:24.809 Say was bouncing and B is going to be be minus, say. 408 01:01:24.809 --> 01:01:29.429 If a reasonable and be in the books between 0 and 1. 409 01:01:29.429 --> 01:01:34.469 Okay, so now if we have event a. 410 01:01:34.469 --> 01:01:43.230 What's what's its profitability going to be? Some just say, type it in a chat so event a is interval point. 2. 2.6. 411 01:01:43.230 --> 01:01:49.289 What's it going to be? You won't have any ideas. 412 01:01:51.269 --> 01:01:55.559 You can type it into chat if you want. 413 01:01:59.460 --> 01:02:04.409 Okay, oops. 414 01:02:06.269 --> 01:02:16.829 Any ideas point 6 minus point 2, right? Which is. 415 01:02:18.329 --> 01:02:22.230 4 B. 416 01:02:26.010 --> 01:02:30.059 And what's the probability of event to be happening? 417 01:02:30.059 --> 01:02:34.230 Oops, that is a 1 point there. Okay. 1. 418 01:02:36.719 --> 01:02:41.550 Okay, any ideas for B, right? It's going to be. 419 01:02:41.550 --> 01:02:46.920 46, now, what we can do is that. 420 01:02:48.210 --> 01:02:59.065 We can now find the probability, and they're probably going to. 421 01:02:59.094 --> 01:03:04.284 Now, you can do this say, we're just looking at the thing um, probability say of. 422 01:03:07.110 --> 01:03:11.760 A union be in this case that's a probability. Basically the interval. 423 01:03:11.760 --> 01:03:18.090 Um, I don't know if I'll say. 424 01:03:18.090 --> 01:03:24.179 Point 2 up to 1 or something that's going to be 8. 425 01:03:24.179 --> 01:03:32.099 And now we can run the formula backward. So we can say. 426 01:03:32.099 --> 01:03:35.550 Probably, they intersect B. 427 01:03:39.510 --> 01:03:46.440 Well, I guess the probability of a B. 428 01:03:46.440 --> 01:03:49.800 We can take the previous form where you work. Good. 429 01:03:49.800 --> 01:03:53.010 And we would get something like, um. 430 01:04:05.460 --> 01:04:10.889 You can do something like that and so on. 431 01:04:13.230 --> 01:04:25.650 Okay, and we have also things like a compliment. So it'll see up there. 432 01:04:27.599 --> 01:04:40.110 That's the compliment of a okay and stuff like that and you can work that out. So. 433 01:04:51.059 --> 01:04:57.510 You know, work with formulas like that, and so on. 434 01:04:57.510 --> 01:05:05.369 Any question, let me work out another example, with the noisy communication, let's say. 435 01:05:18.204 --> 01:05:32.695 Hello. 436 01:05:51.960 --> 01:05:56.579 Same thing as before will we transmit 3 times? 437 01:06:00.360 --> 01:06:07.710 Right. Let me see times. Well, now, let me do the probability of the error say point 1. 438 01:06:09.090 --> 01:06:13.980 Before I had 401 so point 1. 439 01:06:13.980 --> 01:06:19.739 Okay, so that's a probability is a bit going bad. 440 01:06:21.150 --> 01:06:33.900 So, basically, no error. 441 01:06:35.369 --> 01:06:42.719 In the 3 tries well, at point 9 that the 1st bit went through times point 9 and the 2nd bit went through. 442 01:06:42.719 --> 01:06:46.230 Point 9 that the 3rd bit went through. 443 01:06:46.230 --> 01:06:50.820 Which was the trouble with this whiteboard. 444 01:06:52.170 --> 01:06:58.199 Just point 729 probability of 1 error. 445 01:07:00.119 --> 01:07:03.329 Well, it could be the 1st bit could be good. 446 01:07:05.130 --> 01:07:08.789 Times the 2nd, good times the 3rd bit bad. 447 01:07:09.809 --> 01:07:15.599 53 cause it could be which 1 of the 3 pits was bad and the other 2 are good. 448 01:07:15.599 --> 01:07:19.230 And this is point. 449 01:07:23.610 --> 01:07:28.110 The probability of 2 errors, it could be. 450 01:07:28.110 --> 01:07:40.619 Be the 1 good bit of the 3 is either 1 of the 3 so point 9 and a bit was good times. Point 1 the other bit was bad. Touchpoint 1. that's the other bit was bad. 451 01:07:40.619 --> 01:07:46.380 And this is, um, let's see. 452 01:07:48.210 --> 01:07:53.340 Probability of 3 bad fits. 453 01:07:53.340 --> 01:07:58.139 It's going to be the bit is bad point. 1 times point 1. 454 01:07:58.139 --> 01:08:05.880 Point 1, and if I add them up, let's just check to make sure. 455 01:08:05.880 --> 01:08:11.579 729 9 972. 456 01:08:11.579 --> 01:08:16.739 9,909, and that adds up to 1. 457 01:08:22.680 --> 01:08:35.005 Probability of a correct transmission is, 458 01:08:35.005 --> 01:08:35.604 um, 459 01:08:35.635 --> 01:08:41.604 it's that transmission is received correctly if either 0 is received or 1 bit. 460 01:08:41.699 --> 01:08:44.970 If I use 0, bad bits or 1 bad fit. 461 01:08:44.970 --> 01:08:48.810 And that's point 7, 9. 462 01:08:48.810 --> 01:08:55.020 Plus point 2, 4, 3. 463 01:08:56.250 --> 01:08:59.760 Which is point, um. 464 01:08:59.760 --> 01:09:06.630 And 9.972. 465 01:09:10.800 --> 01:09:17.729 So, I just transmit the bit 11 bit. It's a 10% chance of arriving wrong. 466 01:09:17.729 --> 01:09:22.920 If I transmit the 3 bits and vote for best 2 to 3 to see what it should be. 467 01:09:22.920 --> 01:09:27.000 The chats of an error fell 2.28. 468 01:09:27.000 --> 01:09:34.710 So, here the probability of an error or minus point 9. 469 01:09:36.390 --> 01:09:44.850 Equals point, so, here it was an is gonna fell by about a factor of 4. 470 01:09:44.850 --> 01:09:54.300 When do you see how the thing works out? Okay. 471 01:09:57.119 --> 01:10:08.520 And then we can, um, go 1st, any questions on this. 472 01:10:09.960 --> 01:10:22.470 And I'll do 1 more example. Okay. Let me do 1 more quick. 473 01:10:23.819 --> 01:10:29.970 Why do we multiply by 3? Because we're, we're sending 3 bits. 474 01:10:29.970 --> 01:10:44.670 And the error could either be in the 1st transmitted bit, or it could be in a 2nd transmitted fit, or it could be in the 3rd transmitted bed because I want to find the probability of exactly. 1 error of the 3 pits. 475 01:10:48.569 --> 01:10:51.630 There is a formula yeah. 476 01:10:51.630 --> 01:10:58.649 There is a formula and we will see it. Thank you Sam as you're running ahead of me. 477 01:10:58.649 --> 01:11:03.779 It's the choose function or the combination function. Um. 478 01:11:03.779 --> 01:11:07.979 Yeah, just said, let me do it. Yeah, so working ahead. 479 01:11:07.979 --> 01:11:12.029 Um, it would be, um, there's different ways of writing it. 480 01:11:12.029 --> 01:11:15.510 Um, 1 way to write it is. 481 01:11:15.510 --> 01:11:23.279 And choose K and factorial over K factorial. 482 01:11:23.279 --> 01:11:32.430 And minus K factorial. So this is the number of ways you can exactly. You can choose exactly. K items. 483 01:11:32.430 --> 01:11:32.970 From, 484 01:11:32.994 --> 01:11:33.475 and 485 01:11:52.015 --> 01:11:53.305 we'll see this exact, 486 01:11:53.364 --> 01:11:54.505 we'll see this later. 487 01:11:56.220 --> 01:12:00.449 I was working, I just wanted to move up into this slowly, so. 488 01:12:01.590 --> 01:12:08.489 Okay, that's also there's also different ways to write this. It's also written or something like an. 489 01:12:08.489 --> 01:12:23.305 Other notations, and so on choose foreign or the combination formula any other questions. So, for this case here any K equals 1. 490 01:12:23.814 --> 01:12:28.885 so 3 factorial over 1 factorial 2 factorial. 491 01:12:29.130 --> 01:12:32.880 Is 3 over 1. 492 01:12:32.880 --> 01:12:36.659 Well, 3 factorial is 6 and over 1 times 2. 493 01:12:37.979 --> 01:12:41.550 On plus 3. 494 01:12:43.140 --> 01:12:49.800 Okay, any other questions why is the correct transmission? Well. 495 01:12:49.800 --> 01:12:54.630 Well, the receiver is doing the best 2 out of 3 vote. Now. 496 01:12:54.630 --> 01:12:58.319 So, when we transmit the bit 3 times. 497 01:12:58.319 --> 01:13:03.270 And every time we transmit it, there is a 90% chance that will arrive correctly. 498 01:13:03.270 --> 01:13:10.590 So, if we transmit 3 bits, the chance that all 3 arrive correctly is point 9 cute. 499 01:13:10.590 --> 01:13:16.079 And point 9 tube, if I did, my math correctly is point 729. 500 01:13:16.555 --> 01:13:26.185 You can always check me on math. The transmissions received correctly also. Exactly. 2 of the 3 bits arrived correctly. Exactly too. 501 01:13:26.694 --> 01:13:34.104 Um, so that would be if the 1st bits correct and the 2nd batch. Correct but the 3rd page is bad. 502 01:13:34.319 --> 01:13:38.460 That'll be point 9.9 tax point 1. 503 01:13:38.460 --> 01:13:46.590 Which will be point, but it could also be the 1st, bits. Good. The 2nd bit is bad. And the 3rd bid is good. 504 01:13:46.590 --> 01:13:52.470 So, it also be 481 or it could be the 1st, 2 bits are good. And the 3rd bid is bad. 505 01:13:52.470 --> 01:13:58.289 That would also be 181 and 3 times 181 is point. 2 4, 3. 506 01:13:59.430 --> 01:14:13.140 So, we had the probability for all 3 bids arriving correctly to the probability for exactly. 2 bits arriving correctly and that's our 15 but we don't care which of the 3. it's good. It's bad in which which to work right? 507 01:14:17.460 --> 01:14:29.640 Any other questions so that's all the material for today. Um, so Monday's a holiday we'll continue on with, um. 508 01:14:31.020 --> 01:14:39.810 We'll continue on with, um, we're working in chapter 2, so if you want to read ahead, you can be reading in chapter 2. so. 509 01:14:39.810 --> 01:14:45.479 So, I'll stay online for another minute or so. 510 01:14:48.000 --> 01:14:59.010 No, um, you multiply by 3? Um, no. Um, the point 729 is the probability that all 3 bits arrive. Good. 511 01:14:59.010 --> 01:15:03.960 The point 243 combines 3 cases. 512 01:15:03.960 --> 01:15:09.210 The 1st case for the 1st, 2 bits being good and the 3rd fit being bad. 513 01:15:09.210 --> 01:15:18.630 Added to the probability that the bad bit is the 2nd of the 3 and added to the, the bad fits the 3rd bit of the 3. 514 01:15:18.630 --> 01:15:22.380 And each 1 of those 3 cases point 0, 8, 1. 515 01:15:22.380 --> 01:15:26.310 The 3 of them are at the point 2, 4, 3. 516 01:15:28.079 --> 01:15:29.185 Does that make sense? 517 01:15:42.774 --> 01:15:45.505 Right? Exactly. Any 1 of the 3 bits. 518 01:15:45.720 --> 01:15:50.909 But we can do in case by case by case specifically the 1st, bit being bad. 519 01:15:50.909 --> 01:15:53.970 Is 108 1. 520 01:15:53.970 --> 01:16:04.734 Specifically, the 2nd bit being bad is also point 081 specifically the 3rd bit being bad. It's also point 081 cause it seems symmetric. So the probability that exactly. 521 01:16:04.734 --> 01:16:11.425 1 of the 3 fits is bad, but we don't care which 1 that is point. 2 4 3. 522 01:16:20.069 --> 01:16:24.930 I'll just continue on homework 1 and. 523 01:16:26.399 --> 01:16:30.930 Technical easy like, if you think about this, and I'll hit you with more homeworks. 524 01:16:30.930 --> 01:16:32.305 The future so 525 01:16:46.944 --> 01:16:49.045 can you go back to the Venn diagram? 526 01:16:49.465 --> 01:16:52.164 Sure. I'll draw it again. Larger for you. 527 01:17:23.699 --> 01:17:33.239 Okay, so. 528 01:17:34.319 --> 01:17:37.949 Well, let me do the 2 case for us. Okay. Um. 529 01:18:14.970 --> 01:18:28.619 Okay. 530 01:18:30.659 --> 01:18:43.020 So, if we do the 2 thing, a intersect be the problem. So, a N. V. to probably the size of the set can make it the same thing. So it's a plus B minus the thing in the middle that was double counted. 531 01:18:43.020 --> 01:18:46.319 Okay, now 3 thing. 532 01:18:47.640 --> 01:18:55.500 So. 533 01:19:10.409 --> 01:19:13.710 You do it in different colors here let's say so. 534 01:19:13.710 --> 01:19:18.329 You do this this. 535 01:19:18.329 --> 01:19:25.890 This so we double counted and triple counted stuff. So now what we have to do. 536 01:19:26.305 --> 01:19:27.114 Minus, 537 01:19:27.114 --> 01:19:27.564 um, 538 01:19:41.725 --> 01:19:43.854 and that's going to be subtracting. 539 01:19:47.880 --> 01:19:58.409 Okay, so but then, so what we did is we over subtracted the stuff in the middle. 540 01:19:58.409 --> 01:20:05.789 So, okay. 541 01:20:08.939 --> 01:20:16.800 Cause the stuff in the middle we added 3 times, then we subtracted 3 times and we added again. So it got counted once. 542 01:20:16.800 --> 01:20:29.399 So so, the Venn diagram, why is the 3 case intersection? Probably not added back twice. 543 01:20:32.850 --> 01:20:40.590 Well, in the middle here, it's added in 3 times, it's subtracted 3 times and it's added it again. 544 01:20:40.590 --> 01:20:45.090 So, it comes out to 1. does that look okay, Max. 545 01:20:45.090 --> 01:20:55.140 I'll get to the next question after this. Okay. Thanks any more Venn diagram questions. 546 01:21:00.960 --> 01:21:04.949 Okay, that's, um, transmission again. 547 01:21:12.300 --> 01:21:15.989 Okay, so what we're what we're doing here. 548 01:21:17.909 --> 01:21:26.460 Okay, so transmit a bit. 549 01:21:26.460 --> 01:21:36.420 3 times receiver votes. 550 01:21:36.420 --> 01:21:41.550 Okay, for the best 2 out of 3. okay. So, um. 551 01:21:46.229 --> 01:21:53.279 So, the probability that a bit. 552 01:21:53.279 --> 01:22:03.329 Is is bad so it's received bad, let's say, was point 1 or something. Okay. 553 01:22:04.710 --> 01:22:13.710 Okay, so so the probability. 554 01:22:15.119 --> 01:22:23.130 The bid is received good equals point 9. 555 01:22:23.130 --> 01:22:26.609 Okay. 556 01:22:28.079 --> 01:22:34.710 So the probability the bed is good. 557 01:22:34.710 --> 01:22:40.409 3 times 8.9. 558 01:22:40.409 --> 01:22:45.630 Cube trouble with the white board. 559 01:22:50.909 --> 01:23:00.149 729 okay, add 3 times. 560 01:23:02.880 --> 01:23:06.600 His point 1. 561 01:23:08.489 --> 01:23:12.779 Okay, um. 562 01:23:15.239 --> 01:23:18.689 The probability let's say, it's good. 563 01:23:18.689 --> 01:23:22.020 Good bad. 564 01:23:22.020 --> 01:23:25.050 These are the 3 transmission. 565 01:23:27.479 --> 01:23:32.130 Is the 1st transmission the 2nd and the 3rd bid. 566 01:23:35.640 --> 01:23:45.899 Very good there. Okay. Um, so probably the 1st page was good is point 9. 567 01:23:46.920 --> 01:23:53.819 2nd, here, no point 9 to get it's, um, so precise. 568 01:23:56.699 --> 01:24:03.750 Come on 1.9.9 times point 1. 569 01:24:05.819 --> 01:24:10.619 Oh, 81 so far looking good. 570 01:24:14.909 --> 01:24:17.909 Probability of good, bad. Good. 571 01:24:19.439 --> 01:24:26.489 It's also point 081, it's point 9. 9.1. 572 01:24:26.489 --> 01:24:31.949 9.9 no problems here with the white board. 573 01:24:36.390 --> 01:24:39.479 1.9.1. 574 01:24:39.479 --> 01:24:44.430 Point 9 point. 575 01:24:44.430 --> 01:24:49.619 81, and so on the probability of. 576 01:24:51.750 --> 01:24:57.449 Good good good point. 1. 577 01:24:57.449 --> 01:25:03.029 Point 9.9.81. 578 01:25:03.029 --> 01:25:09.600 And so on, okay, if you can read the handwriting, like. 579 01:25:10.619 --> 01:25:14.760 Does that make sense? 580 01:25:16.140 --> 01:25:24.750 So, the probability of exactly 1, bad fit is point 081 time 3, which is point 2, 4, 3. 581 01:25:26.039 --> 01:25:35.340 That makes sense. 582 01:25:47.069 --> 01:26:00.000 Okay. 583 01:26:04.920 --> 01:26:10.500 We're adding the, we're adding the times 3 because what we're doing. 584 01:26:12.659 --> 01:26:16.260 Is we're adding this plus this. 585 01:26:16.260 --> 01:26:19.380 What's this? That gives us. 586 01:26:31.614 --> 01:26:46.585 Any other questions think about it for a minute you leave if you want, we're past the end of the class time. I'll stay around for another. Um. 587 01:26:46.949 --> 01:26:57.119 I'll stay around for 1 more minute, um, to give people a chance to think of any questions and it says no, no more question style and so. 588 01:26:59.399 --> 01:27:02.399 Feel free to call and moving, so. 589 01:27:10.920 --> 01:27:19.649 Yeah, thank you. Tyler. Okay, thank you. 590 01:27:25.500 --> 01:27:34.319 Okay, the only question you have now is. 591 01:27:38.939 --> 01:27:43.380 Is still the correct trend? I don't understand your question. 592 01:27:50.729 --> 01:27:54.210 Professor, so the correct transmission is. 593 01:27:54.210 --> 01:27:59.850 Just a SEC, I've got your volume down to low here. Okay. Could you say it again? Please. 594 01:27:59.850 --> 01:28:09.060 Professors are in the previous note, it says the probability of correct transmission is point 7 to 9. plus. 595 01:28:09.060 --> 01:28:14.369 Point 243, so yeah, that's correct. I add this to together. 596 01:28:14.369 --> 01:28:20.039 Okay, because there's different ways the transmission can be correct. 597 01:28:20.039 --> 01:28:31.409 Just a 2nd here um, yeah, there's different ways that the transmission can be. Correct. The 1st way is that all 3 bits are received correctly. 598 01:28:31.409 --> 01:28:38.220 The 2nd way is that any 2 bits I received correctly? And the 3rd bid is wrong. 599 01:28:39.270 --> 01:28:46.770 So, the 1st way is, the probability is point 729 that all 3 bits are received correctly the 2nd way that. 600 01:28:46.770 --> 01:29:00.175 2 bits I received correctly and 1 bit is wrong. We don't care which of the 3 is wrong. That's the point 243. so we add those 2 cases cause they're discharged. They can't both occur. I mean, 1 possibility is all 3 bits arrive. Okay. 601 01:29:00.204 --> 01:29:11.574 The 2nd possibility is 2 of them arrive. Okay. And the 3rd is wrong and we add those 2 cases, because in both those cases, the receiver will receive a bit correctly. 602 01:29:12.750 --> 01:29:19.500 But if 2 bits are changed in the transmission, then the receiver will guess the wrong bit. 603 01:29:22.079 --> 01:29:34.260 Does that make sense? Um, so do you mean, like, so because. 604 01:29:34.260 --> 01:29:45.899 We add this to together because this is a 2nd, I can hardly hear you. I'm going to try and put my volume Max, but. 605 01:29:45.899 --> 01:29:54.869 Um, my volume is basically. 606 01:29:57.779 --> 01:30:02.489 Um, my volume's high, but. 607 01:30:02.489 --> 01:30:05.939 Could you tell me your question again? Please. 608 01:30:05.939 --> 01:30:10.680 So, professor, so so you add this to. 609 01:30:10.680 --> 01:30:16.859 Together is that so the point 203 is like. 610 01:30:16.859 --> 01:30:19.949 Only only 1. 611 01:30:19.949 --> 01:30:23.399 1 day get wrong, right? 612 01:30:25.979 --> 01:30:28.979 Yeah, let me type it. I mean, you know, um. 613 01:30:42.420 --> 01:30:53.699 So, for the point, 027 is like, 22 gets wrong. 614 01:30:55.949 --> 01:31:08.069 Okay, does that help there? Um, that helps, but I still don't get. Why do you add this to together? You don't add the 0 point. 615 01:31:08.069 --> 01:31:16.439 Um, 0277, because that's the probability that 2 beds. W. W. W what I want. 616 01:31:18.510 --> 01:31:21.810 Is the problem that the receiver. 617 01:31:24.000 --> 01:31:29.819 Guess it's correct. Um. 618 01:31:32.130 --> 01:31:38.039 And that happens if 3 or 2. 619 01:31:40.079 --> 01:31:46.140 Okay, if 1, only 1 bit. 620 01:31:49.050 --> 01:31:52.140 Okay, the receiver. 621 01:31:52.140 --> 01:32:02.099 I guess is wrong so it's the correct transmission when it's all 3 bits or if it's exactly to. 622 01:32:02.844 --> 01:32:03.354 Correct. 623 01:32:03.413 --> 01:32:20.873 Okay. 624 01:32:24.238 --> 01:32:28.798 Oh, that really helps. Thank you so much for professor. I. okay. 625 01:32:30.359 --> 01:32:38.399 Correct Tyler? Yeah. Yes. 626 01:32:49.889 --> 01:32:55.588 Thank you professor have a good day. Okay have a good weekend. Any other questions. 627 01:32:55.588 --> 01:33:01.918 So, if no more questions. 628 01:33:01.918 --> 01:33:06.658 Thank you professor. You're welcome. Okay. Okay. 629 01:33:17.429 --> 01:33:18.599 Okay.