WEBVTT 1 00:00:06.929 --> 00:00:10.528 Okay. 2 00:00:10.528 --> 00:00:15.599 So, what I'm trying to do is record on Webex. 3 00:00:15.599 --> 00:00:19.708 So. 4 00:00:23.640 --> 00:00:27.660 Right. 5 00:00:40.320 --> 00:00:44.520 Hello. 6 00:00:46.109 --> 00:00:52.170 Right. 7 00:00:52.170 --> 00:00:57.990 Okay. 8 00:01:11.579 --> 00:01:15.329 Huh. 9 00:01:18.900 --> 00:01:23.430 And. 10 00:01:23.430 --> 00:01:30.239 Cool. Okay so they're actually started working. 11 00:01:30.239 --> 00:01:33.510 It's probability probably. 12 00:01:33.510 --> 00:01:40.140 And. 13 00:01:40.140 --> 00:01:44.969 Okay. 14 00:01:44.969 --> 00:01:51.209 Huh. 15 00:01:51.209 --> 00:01:54.900 Wow. 16 00:01:57.959 --> 00:02:01.590 Cool. 17 00:02:10.710 --> 00:02:15.300 Hello. 18 00:02:15.300 --> 00:02:18.539 Oh, okay. 19 00:02:20.699 --> 00:02:26.939 ta's have office hours posted here the host them via a Webex. 20 00:02:26.939 --> 00:02:37.530 So that can join and if no, 1 joins within 15 minutes, they'll go away whoever you want other times, right? There. 21 00:02:38.909 --> 00:02:46.860 And so we'll have 1 Webex for the foreseeable future people preferred for the Webex is easier. I think. 22 00:02:47.909 --> 00:02:57.210 And what I'll do for lectures for the foreseeable future is, I'll lecture here in person, and I love to broadcasted on Webex. 23 00:02:57.210 --> 00:03:02.610 And also what I write down, I upload as a PDF. 24 00:03:02.610 --> 00:03:08.189 So, and this technology fails sometimes, but, you know. 25 00:03:08.189 --> 00:03:16.020 Work and, of course, just to remind you, you go up here in the files tab. 26 00:03:16.020 --> 00:03:19.500 There are some things. Okay. 27 00:03:19.500 --> 00:03:25.439 The digits of the front, or the class number. Okay. Next thing. Um. 28 00:03:27.419 --> 00:03:31.740 We need to have a midterm job. I have 2 midterm times in a final. 29 00:03:31.740 --> 00:03:38.939 And Feb, 24th or 28th I'm open to either 1. 30 00:03:38.939 --> 00:03:44.789 Open the class for if you have a strong opinion 1 way, or the other. 31 00:03:46.229 --> 00:03:50.669 Coming out yeah. 32 00:03:50.669 --> 00:03:54.870 No exam at all is not an option 1 exam. 33 00:03:54.870 --> 00:03:58.439 You prefer. 34 00:03:58.439 --> 00:04:02.460 Well, you can think about it and. 35 00:04:02.460 --> 00:04:05.520 And I may ask again, Thursday. 36 00:04:05.520 --> 00:04:10.650 Will be in class. Okay. And. 37 00:04:10.650 --> 00:04:16.860 People who are quarantine I don't know what we'll do. Um, so that, um. 38 00:04:16.860 --> 00:04:23.310 I have some stuff here. I got to edit the stuff I've typed in reviewing what I've done for people. 39 00:04:23.310 --> 00:04:31.288 That what I'll do today is I want to then come back circle back with some chapter 2 stuff and get some chapter 3 stuff. 40 00:04:31.288 --> 00:04:38.218 Probability in the real world. This is a cool 1. 41 00:04:41.218 --> 00:04:50.639 It's gone, um, but this was everyone knows what it is. Okay so. 42 00:04:50.639 --> 00:04:55.829 They proposed having a future's market on current events, like. 43 00:04:55.829 --> 00:04:59.608 You know, what's the probability of war breaking out? 44 00:04:59.608 --> 00:05:07.978 No, wherever, and they would let people buy options and it basically, and they think that the price. 45 00:05:07.978 --> 00:05:12.749 Of the option might reflect how would actually happening. Um. 46 00:05:12.749 --> 00:05:18.269 Got a little bad press, and it became public. So they brought the idea. Um. 47 00:05:18.269 --> 00:05:22.228 I think the concept is that there's a moral hazard. 48 00:05:22.228 --> 00:05:32.158 That, you know, somebody who gets the kingdom of Albania to pick a pixel example, it's going to get assassinated and puts a lot of money on that. 49 00:05:32.158 --> 00:05:41.038 Like, then, you know, be inclined to go and assassinate the king of operating himself and something, so it can collect. So okay. Um. 50 00:05:41.038 --> 00:05:45.059 Mentioned a real king of Albania that had. 51 00:05:45.059 --> 00:05:49.168 5,755 assassination attempts. 52 00:05:49.168 --> 00:05:54.119 Can't do anything, right? No. Okay. Um. 53 00:05:54.119 --> 00:05:59.728 Okay, I'll walk through this example before, um, some of stuff I typed in. 54 00:05:59.728 --> 00:06:02.908 Total probability events, um. 55 00:06:02.908 --> 00:06:10.858 And I'll hit face, so you can read this there. I have to edit this and so on. But the idea. 56 00:06:10.858 --> 00:06:14.278 Okay, then I'll get into some chapter for you. 57 00:06:17.668 --> 00:06:21.598 Okay. 58 00:06:21.598 --> 00:06:28.168 Okay. 59 00:06:28.168 --> 00:06:31.678 Okay, um. 60 00:06:35.399 --> 00:06:39.509 Hello. 61 00:06:46.199 --> 00:06:53.309 There's some conditional stuff. These numbers are fictitious. Of course. 62 00:07:06.569 --> 00:07:13.439 Huh. 63 00:07:13.439 --> 00:07:16.649 Let me. 64 00:07:16.649 --> 00:07:19.649 I just want to go through and pick up this. 65 00:07:19.649 --> 00:07:24.778 For example. 66 00:07:24.778 --> 00:07:28.319 Engineers underground. Okay. 67 00:07:30.059 --> 00:07:35.488 So. 68 00:07:38.428 --> 00:07:41.999 Hello. 69 00:07:44.428 --> 00:07:47.999 So, basically we were talking. 70 00:07:49.709 --> 00:07:54.838 Fictional University. 71 00:07:54.838 --> 00:08:01.139 And we might have. 72 00:08:01.139 --> 00:08:05.399 Is their engineers, the undergrad. 73 00:08:07.588 --> 00:08:11.309 And who knows what we might have. 74 00:08:11.309 --> 00:08:15.478 Is, um. 75 00:08:23.069 --> 00:08:30.209 Okay, so the total number is. 76 00:08:32.068 --> 00:08:35.639 Plus, 1, plus 2, plus 1, 4. 77 00:08:35.639 --> 00:08:40.168 Do a little this before it's this I'm going to review in more detail. 78 00:08:41.908 --> 00:08:45.899 The probability of a student is an undergrad of courses. 3 7. 79 00:08:45.899 --> 00:08:49.379 Oh, Bill probably going to probably even undergrad is. 80 00:08:50.729 --> 00:08:54.658 I Canada. 37247. 81 00:08:54.658 --> 00:09:02.339 Undergrad 37 now, we can start talking about probability of an engineer, given you an undergrad. 82 00:09:02.339 --> 00:09:07.769 Is the undergraduate 3001 of their engineers 1 or. 83 00:09:07.769 --> 00:09:11.038 I did this before, but it had a little confusion. 84 00:09:11.038 --> 00:09:15.839 And we do the opposite, the probability that you're an undergrad, given you an engineer. 85 00:09:15.839 --> 00:09:24.119 If the 4,000 engineers, 1000 like that. So I'll look at the conditional probability. 86 00:09:24.119 --> 00:09:27.479 Um. 87 00:09:27.479 --> 00:09:32.818 And now we might want to give it as an example for base here. Exactly. 88 00:09:32.818 --> 00:09:36.808 Um, so. 89 00:09:40.408 --> 00:09:45.418 Okay. 90 00:09:49.889 --> 00:09:56.698 And it's a way to go the other direction on conditional probability. So. 91 00:10:06.298 --> 00:10:09.328 You might say backwards conditional probability. 92 00:10:11.938 --> 00:10:22.109 Let me show me. 93 00:10:22.109 --> 00:10:25.769 Is. 94 00:10:25.769 --> 00:10:28.948 I do total Internet. 95 00:10:28.948 --> 00:10:32.729 Okay. 96 00:10:32.729 --> 00:10:37.198 Okay. 97 00:10:37.198 --> 00:10:40.619 Oh. 98 00:10:40.619 --> 00:10:44.158 Perfect. 99 00:10:44.158 --> 00:10:48.298 Hello. 100 00:10:49.708 --> 00:10:55.918 You want to find probably engine engineering side given you're an undergrad. 101 00:10:55.918 --> 00:11:03.389 And how can we do that? Well, we know that the probability of being an engineer and an undergrad. 102 00:11:03.389 --> 00:11:07.948 Is the probability of engineering given undergrad type of probability of undergrad? 103 00:11:07.948 --> 00:11:13.589 And which is also the probability of an undergrad given engineer and kinds of probability of that. 104 00:11:13.589 --> 00:11:16.678 Okay, so, um. 105 00:11:16.678 --> 00:11:20.639 And we also know that. 106 00:11:20.639 --> 00:11:27.839 The probability I'll say of being an undergrad. 107 00:11:29.578 --> 00:11:36.928 This will be a new thing is the probability of an undergrad and engineering plus. 108 00:11:36.928 --> 00:11:41.458 Probability of undergrad and not engineering. 109 00:11:41.458 --> 00:11:47.548 Okay, this is a this is getting into a total probability idea here. 110 00:11:52.589 --> 00:11:55.979 Um. 111 00:11:55.979 --> 00:11:59.519 So, what we can do is that. 112 00:11:59.519 --> 00:12:02.818 Um. 113 00:12:02.818 --> 00:12:06.538 We can say the probability of. 114 00:12:06.538 --> 00:12:11.578 Hello. 115 00:12:14.729 --> 00:12:18.839 An undergrad divided by. 116 00:12:18.839 --> 00:12:23.428 Probability of undergrad. 117 00:12:48.899 --> 00:12:53.999 Not engineering undergrad let's say. 118 00:12:53.999 --> 00:12:58.619 Now, the probability of engineering and undergrad. 119 00:13:00.359 --> 00:13:04.109 We can also write that down as the probability of of. 120 00:13:05.999 --> 00:13:11.099 That are up there. 121 00:13:11.099 --> 00:13:19.139 And the probability is not engineering given undergrad and undergrad. 122 00:13:19.139 --> 00:13:22.379 Probability of and undergrad. 123 00:13:22.379 --> 00:13:27.208 Probably undergrad given not engineering kinds of problems if not engineering. 124 00:13:27.208 --> 00:13:30.509 We can put the whole thing together. 125 00:13:30.509 --> 00:13:43.678 Equals. 126 00:13:53.969 --> 00:13:58.649 Hello. 127 00:14:05.068 --> 00:14:09.688 Yeah, we can start doing stuff like that. So, what this means is this, that. 128 00:14:09.688 --> 00:14:20.009 We have things like the probability of an undergrad if you're an engineer and we want to find the offices. Yes. 129 00:14:20.009 --> 00:14:24.688 How are you going to call. 130 00:14:24.688 --> 00:14:27.839 Profitability of. 131 00:14:27.839 --> 00:14:31.168 Um. 132 00:14:31.168 --> 00:14:37.708 So, we're here here here in the middle 1 here. 133 00:14:37.708 --> 00:14:46.859 It looks like you have a probability on the opposite. 134 00:14:46.859 --> 00:14:55.139 And not engineering. Oh, sorry. Let me write it. You got got picked up by my messy handwriting. 135 00:14:56.818 --> 00:15:00.749 Hello. 136 00:15:02.129 --> 00:15:06.928 Yeah, so. 137 00:15:06.928 --> 00:15:10.379 So, we can start doing stuff like this. So. 138 00:15:10.379 --> 00:15:13.828 What we have. 139 00:15:13.828 --> 00:15:22.048 Are probabilities of under undergrad, given your engineering, and we want to go backwards and find the probability. 140 00:15:22.048 --> 00:15:25.499 Of engineering, given undergrad and. 141 00:15:25.499 --> 00:15:29.759 The example, here let's say, is that. 142 00:15:29.759 --> 00:15:32.788 If I scroll back up. 143 00:15:34.708 --> 00:15:40.379 Let me see, probability of engineering is 47, but down here. 144 00:15:40.379 --> 00:15:50.668 And again, let me write this more neatly for you. 145 00:15:54.359 --> 00:15:58.499 This is engineering. 146 00:16:05.399 --> 00:16:12.058 Okay, and so probably been engineering was 47. 147 00:16:12.058 --> 00:16:18.028 I probably we've undergrad given engineering if I can scroll back here. 148 00:16:18.028 --> 00:16:21.269 Undergrad given your engineering. 149 00:16:21.269 --> 00:16:27.208 Is 1 quarter and undergrad given your noggin. 150 00:16:27.208 --> 00:16:30.599 Undergrad given your engineering is 1 a quarter. 151 00:16:33.089 --> 00:16:36.269 Undergrad giving you not an engineer. 152 00:16:36.269 --> 00:16:39.749 Is if you outside. 153 00:16:39.749 --> 00:16:45.239 There's 3000 kids in 2000 of those are undergrads. So that's 2 thirds. 154 00:16:49.438 --> 00:16:53.188 And it's all nets up to. 155 00:16:53.188 --> 00:16:57.119 1, 7. 156 00:16:58.168 --> 00:17:01.198 The bottom is 1 7th plus. 157 00:17:01.198 --> 00:17:06.689 3. 158 00:17:08.818 --> 00:17:14.368 Probably being an engineering, if you're an undergrad, let me go back up and check that. 159 00:17:15.598 --> 00:17:18.659 That is correct. So. 160 00:17:18.659 --> 00:17:23.249 So, basically we had some. 161 00:17:23.249 --> 00:17:32.159 What we had here in words is we had we knew the probability with you're an engineer at T. V. and we knew the probability of. 162 00:17:32.159 --> 00:17:40.409 Being an undergrad, if you're an engineer and undergrads not and then we converted it to find the probability that you're an engineer. 163 00:17:40.409 --> 00:17:52.648 If you're an undergrad, it takes some thinking so, let me do another example or 2 3. 164 00:17:52.648 --> 00:18:00.088 I did examples before with communication, so the. 165 00:18:00.088 --> 00:18:03.659 Wow. 166 00:18:03.659 --> 00:18:11.548 Hello. 167 00:18:11.548 --> 00:18:16.288 This is another another example. 168 00:18:22.078 --> 00:18:25.588 Um. 169 00:18:29.489 --> 00:18:32.848 Well, the relevant things are we going to get people. 170 00:18:34.019 --> 00:18:42.479 We're going to looking for lt, Jane. 171 00:18:42.479 --> 00:18:48.689 And there's probably a misspell and looking for a, and we have a 10. 172 00:18:48.689 --> 00:18:53.308 Okay, so let's say that. 173 00:18:53.308 --> 00:18:56.338 Um. 174 00:18:56.338 --> 00:19:01.499 The probability of having of being aware world is. 175 00:19:01.499 --> 00:19:05.338 I don't know, um. 176 00:19:05.338 --> 00:19:09.989 1, and the 1 at a 100 people are wearables. 177 00:19:09.989 --> 00:19:13.138 Okay, um. 178 00:19:13.138 --> 00:19:19.919 And let's say, and then we're going to call the check this key positive. 179 00:19:19.919 --> 00:19:25.528 So called key and oh, okay. And then so the probability. 180 00:19:25.528 --> 00:19:32.159 Um, the test being positive, if you are aware, will. 181 00:19:33.868 --> 00:19:40.679 Is I don't know, um, 3 quarters. 182 00:19:40.679 --> 00:19:43.888 Doesn't catch them all captured most of them. 183 00:19:43.888 --> 00:19:46.919 But the test might be positive. 184 00:19:46.919 --> 00:19:52.019 The thing is the test that sometimes pause it because if you're not aware of. 185 00:19:52.019 --> 00:19:55.439 And maybe. 186 00:19:55.439 --> 00:20:04.499 I don't know 10 for the time that the test of the false positives. So this would be what would be called, like, a false positive or something. 187 00:20:08.398 --> 00:20:19.078 Okay, so we have so somebody a random person test positive for the wearable key. 188 00:20:19.078 --> 00:20:23.189 Then we want to know that if the test was positive. 189 00:20:23.189 --> 00:20:34.409 What's the probability that you really are aware of? Okay. So if the test was positive, what's the probability that you are aware of? Okay. 190 00:20:36.148 --> 00:20:47.669 So, how can we do that? And you see what we've got the probability from testing falls if you are, and are not aware of what we want the problem is there where it won't if the test is positive. Okay. 191 00:20:47.669 --> 00:20:51.568 So, we can start having fun with this, um. 192 00:20:52.828 --> 00:20:59.969 I see now, um, well, the 1st question is. 193 00:20:59.969 --> 00:21:05.368 How many tests are positive. 194 00:21:09.118 --> 00:21:13.679 Okay, well the probability of a positive test. 195 00:21:13.679 --> 00:21:20.308 This will be total probability the probability of a positive test and you're aware of I'm not going to follow a little. 196 00:21:20.308 --> 00:21:26.159 Intersect side, plus the probability the test is positive that you're not aware. 197 00:21:26.159 --> 00:21:34.828 Okay, probability to test as positive and you're aware of. 198 00:21:36.749 --> 00:21:40.138 Is the probability the positive. 199 00:21:40.138 --> 00:21:43.679 Given your where we'll have the possibility or aware will. 200 00:21:43.679 --> 00:21:46.858 Which is. 201 00:21:50.669 --> 00:21:54.598 3 quarters times 1, 100. 202 00:21:54.598 --> 00:22:02.759 For 100 the probabilities that the test is positive and you're not aware. Well. 203 00:22:02.759 --> 00:22:08.909 And it'll probably be the test is positive given you not aware of. 204 00:22:08.909 --> 00:22:12.239 Well, which is 110th. 205 00:22:12.239 --> 00:22:15.419 99,100. 206 00:22:15.419 --> 00:22:19.739 Not all the. 207 00:22:19.739 --> 00:22:24.959 Okay, so the problem will do the positive. 208 00:22:24.959 --> 00:22:27.989 Some of those. 209 00:22:27.989 --> 00:22:31.378 Hello. 210 00:22:31.378 --> 00:22:37.828 Which is, um. 211 00:22:37.828 --> 00:22:44.009 Um. 212 00:22:50.038 --> 00:22:55.409 Okay. 213 00:22:55.409 --> 00:23:02.128 You can check on that and so now the, what more do I need here? 214 00:23:02.128 --> 00:23:13.618 Build the, I'm aware, well, if the test is positive. 215 00:23:13.618 --> 00:23:17.519 Problem loyalty that I'm both divided by the top. 216 00:23:17.519 --> 00:23:24.088 That which is. 217 00:23:40.138 --> 00:23:44.878 I think feel free to come back to me on that. Um. 218 00:23:45.989 --> 00:23:52.499 Okay, so about 7%, um. 219 00:23:56.068 --> 00:24:02.189 Again, I may have made some enormous math there because I'm doing this medical league. 220 00:24:02.189 --> 00:24:08.459 Um, he checked me on that, so this is the example base role. 221 00:24:08.459 --> 00:24:11.699 That I had the conditionals going in 1 direction. 222 00:24:11.699 --> 00:24:16.199 So, I'll go to the test was positive if I was and was not aware. Well. 223 00:24:16.199 --> 00:24:20.788 I want it the other way I'll build it, I'm aware of is the test positive. 224 00:24:22.499 --> 00:24:26.159 Now, what this says in English is that. 225 00:24:26.159 --> 00:24:30.868 Being aware Wallsten unlikely event. I said it was 1 in a 100. 226 00:24:30.868 --> 00:24:35.038 Okay, tell me where we're looking for. 227 00:24:35.038 --> 00:24:44.969 And and so, so it's called the prior probability that I'm aware of is 1, and a 100. 228 00:24:44.969 --> 00:24:52.229 So, if I do the test, and it's positive in the posterior probability is 7%. What else? 229 00:24:52.229 --> 00:24:56.878 So really increase the probability to to raise it up to 100. 230 00:24:56.878 --> 00:25:01.318 Because the test is there. 231 00:25:01.318 --> 00:25:05.009 Let me write down some of this in English then. 232 00:25:11.459 --> 00:25:21.868 Hello. 233 00:25:21.868 --> 00:25:25.169 That was 1%. 234 00:25:25.169 --> 00:25:36.479 Posterior probability um. 235 00:25:36.479 --> 00:25:40.019 Because so so. 236 00:25:44.128 --> 00:25:49.919 The positive test, um, raises, um. 237 00:25:51.628 --> 00:25:55.769 The probability. 238 00:25:57.868 --> 00:26:02.669 They're not always. 239 00:26:02.669 --> 00:26:06.179 Hello. 240 00:26:06.179 --> 00:26:11.368 250, because. 241 00:26:11.368 --> 00:26:14.788 Hi. 242 00:26:14.788 --> 00:26:19.138 Likely. 243 00:26:19.138 --> 00:26:23.459 It's the only approximate so. 244 00:26:28.888 --> 00:26:32.009 Okay, so there you go. 245 00:26:32.009 --> 00:26:38.429 Yeah. 246 00:26:42.239 --> 00:26:47.278 Hello. 247 00:26:47.278 --> 00:26:52.618 So, this thing, um. 248 00:26:52.618 --> 00:27:00.328 And again, just remind you examples that I gave before, any sort of noisy communications travel. 249 00:27:00.328 --> 00:27:07.469 Questions about this. Would you like another example you get the concept. 250 00:27:07.469 --> 00:27:17.128 It happens, this occurs all the time in engineering. Like, you receive a setting that you want to know what technology most likely transmitted. 251 00:27:17.128 --> 00:27:22.048 And the prior probabilities in the real world are off and not 50, 50. 252 00:27:22.048 --> 00:27:26.159 No. 253 00:27:27.778 --> 00:27:31.019 They're not there we do another thing. 254 00:27:31.019 --> 00:27:34.769 Other example. 255 00:27:37.888 --> 00:27:42.598 Let me say. 256 00:27:42.598 --> 00:27:46.259 Hello. 257 00:27:46.259 --> 00:27:49.798 The domain, and I'll try to create an example on the fly. 258 00:27:49.798 --> 00:27:56.128 What topic would you like the example to be a. 259 00:27:56.128 --> 00:28:01.469 Who is it? What are you. 260 00:28:01.469 --> 00:28:04.709 Sure. 261 00:28:04.709 --> 00:28:08.578 Hello. 262 00:28:11.398 --> 00:28:18.358 Let me do a gambling thing. Okay. I. 263 00:28:20.098 --> 00:28:26.759 Hello. 264 00:28:27.929 --> 00:28:32.009 Hello. 265 00:28:32.009 --> 00:28:36.058 Okay. 266 00:28:36.058 --> 00:28:39.388 Not quite lottery, but, um. 267 00:28:40.979 --> 00:28:45.028 Um. 268 00:28:46.229 --> 00:28:50.368 Okay. 269 00:28:50.368 --> 00:28:56.249 We want total. 270 00:28:57.868 --> 00:29:01.558 And I've got a various events is, um. 271 00:29:12.808 --> 00:29:20.338 For you. 272 00:29:23.368 --> 00:29:27.868 That's a very good point. Okay. Um. 273 00:29:29.189 --> 00:29:37.019 And so the key is. 274 00:29:38.729 --> 00:29:42.209 November. 275 00:29:42.209 --> 00:29:45.838 Okay. 276 00:29:48.209 --> 00:29:51.838 Say, if the. 277 00:29:53.818 --> 00:29:57.479 1st coin. 278 00:29:57.479 --> 00:30:02.578 But I may not be able to. 279 00:30:02.578 --> 00:30:05.788 Okay. 280 00:30:07.259 --> 00:30:16.469 Hello. 281 00:30:20.459 --> 00:30:23.489 Hello. 282 00:30:23.489 --> 00:30:26.969 Probably today is 1 half. Of course. 283 00:30:26.969 --> 00:30:34.588 Um. 284 00:30:34.588 --> 00:30:38.519 Hello. 285 00:30:38.519 --> 00:30:49.739 I feel a little table here. They could be there 1. he could be there or 1. 286 00:30:49.739 --> 00:30:53.969 Inside here was key 0, 1, 1 and 2. 287 00:30:53.969 --> 00:30:58.528 Okay, um, for cases. 288 00:31:00.298 --> 00:31:03.689 But he is specifically total number of heads with 1. 289 00:31:03.689 --> 00:31:12.568 So, that will be here probability of a, and he. 290 00:31:12.568 --> 00:31:17.159 4 days that is the order. 291 00:31:19.528 --> 00:31:29.278 To be given a, the probability of. 292 00:31:29.278 --> 00:31:32.999 By your probability of day. 293 00:31:34.888 --> 00:31:40.798 So, the 1st coin, was that what this means in English it's the 1st point was a head. 294 00:31:40.798 --> 00:31:46.439 It's a 50 50 chance that the 2nd coin is a tail, which means my total is 1. 295 00:31:46.439 --> 00:31:51.118 The fact that my head is 1. okay. What I want. 296 00:31:52.648 --> 00:31:58.259 The probability of if the total number of heads with 1. 297 00:31:58.259 --> 00:32:02.429 What's the probability of the 1st calling with the heck. 298 00:32:03.568 --> 00:32:06.749 Well. 299 00:32:06.749 --> 00:32:12.868 So, the key terms of how will the key goes the. 300 00:32:12.868 --> 00:32:19.048 And. 301 00:32:19.048 --> 00:32:27.058 So, the probability of he given a, it's a half. 302 00:32:28.199 --> 00:32:31.919 For 1 here, the probability of. 303 00:32:35.249 --> 00:32:39.929 So like that 1. 304 00:32:51.269 --> 00:32:57.808 Which is the problem today and we can go backwards is the probability of a. 305 00:32:59.878 --> 00:33:06.959 The key. 306 00:33:07.979 --> 00:33:14.308 Oh, did he given a kind of a photo of a. 307 00:33:14.308 --> 00:33:22.199 Divided by a clock, the probability of. 308 00:33:22.199 --> 00:33:26.578 He was not a problem. 309 00:33:26.578 --> 00:33:30.449 Okay. 310 00:33:31.709 --> 00:33:34.828 So the probability they have. 311 00:33:37.588 --> 00:33:44.818 You probably T, given a was 1 half. 312 00:33:47.729 --> 00:33:51.298 So, we need the probability of T. V. but not a. 313 00:33:52.558 --> 00:33:58.409 So, we have that the probability if they, if not a, then that is, uh. 314 00:33:59.788 --> 00:34:03.538 The fear and. 315 00:34:03.538 --> 00:34:08.878 That'll be a half, so. 316 00:34:10.168 --> 00:34:13.918 And that all net up to 1. 317 00:34:17.489 --> 00:34:22.528 Is the number is worth the best here, but the probability of of. 318 00:34:22.528 --> 00:34:31.528 That the 1st coin with the head, given that the total number of that was 1. 319 00:34:31.528 --> 00:34:35.728 With the half. 320 00:34:37.469 --> 00:34:41.458 That. 321 00:34:41.458 --> 00:34:45.809 Let's do another 1 on Thursday. 322 00:34:45.809 --> 00:34:49.259 Hello. 323 00:34:51.958 --> 00:34:56.398 So, we have expected we have now what we have. 324 00:34:58.228 --> 00:35:06.179 And in chapter 3, that tends to be discrete random variables. 325 00:35:06.179 --> 00:35:10.378 The finite accountable and we've got some function. 326 00:35:12.688 --> 00:35:18.838 Of the random fair RV, a random variable in this context. 327 00:35:18.838 --> 00:35:27.119 And we saw, for example, the expected value of. 328 00:35:27.119 --> 00:35:30.539 That's an RV so capital. 329 00:35:32.579 --> 00:35:37.619 It is the random variable. 330 00:35:37.619 --> 00:35:40.978 4 K. 331 00:35:40.978 --> 00:35:46.349 That's the value. 332 00:35:46.349 --> 00:35:52.139 So, we saw, for example, we should expect the value. 333 00:35:56.188 --> 00:36:01.528 It's also known as the average or whatever. 334 00:36:05.398 --> 00:36:11.458 And if we got some, and basically, that would be. 335 00:36:11.458 --> 00:36:15.449 The sum of all the X, the probability of that. 336 00:36:15.449 --> 00:36:20.278 So, on. 337 00:36:20.278 --> 00:36:29.188 So so your experiment is past the coin. 338 00:36:31.378 --> 00:36:40.318 1, if I had. 339 00:36:42.449 --> 00:36:46.048 And then expect the value back. 340 00:36:46.048 --> 00:36:49.168 Would be. 341 00:36:49.168 --> 00:36:57.028 The probability of seeing a 1 on 1, with the probability of staying is there a stereotype of probability? 342 00:36:57.028 --> 00:37:02.309 So, fair point, not there. 343 00:37:02.309 --> 00:37:05.518 Okay, that's just the review of last time. Um. 344 00:37:06.809 --> 00:37:12.869 And I did it last time, I did the, um, uniform and so on um. 345 00:37:12.869 --> 00:37:18.449 Now, we can get a function of random variables. You may want to. 346 00:37:18.449 --> 00:37:26.849 Um, function. 347 00:37:29.248 --> 00:37:34.528 Of the random variables, like, maybe say you have to die. 348 00:37:36.148 --> 00:37:41.039 And they pay off. 349 00:37:41.039 --> 00:37:44.938 Is I don't know is that, um. 350 00:37:48.478 --> 00:37:55.858 12 and 304, you get paid off 15, you get paid off 2 and. 351 00:37:55.858 --> 00:38:00.028 So, you get paid off for, let's say. 352 00:38:01.409 --> 00:38:06.418 Okay, so what's the expected value? 353 00:38:08.728 --> 00:38:12.150 And it's a, it's a paradigm. Okay. 354 00:38:15.210 --> 00:38:18.960 So, it's, um. 355 00:38:18.960 --> 00:38:25.349 It's 16 times. 016 times 0 plus 1 6 times 0. 356 00:38:25.349 --> 00:38:28.559 Plus 1, 6th time 1. 357 00:38:28.559 --> 00:38:33.480 Plus, 16 times, 2, plus 1, 6 times 4. 358 00:38:33.480 --> 00:38:36.659 And that will be 7, 6. 359 00:38:38.489 --> 00:38:42.449 So, you playing a game here you talk to paradise. 360 00:38:42.449 --> 00:38:47.039 And then what comes up on top is 106. 361 00:38:47.039 --> 00:38:51.420 And you get paid off according to this table here. So this is all. 362 00:38:54.059 --> 00:38:57.239 Okay, 12 and 3 you lose. 363 00:38:57.239 --> 00:39:07.050 For if the topic for comes up on top, you can pay off 1 or 260 paid off for they expect to pay off is 76. so. 364 00:39:07.050 --> 00:39:14.519 The random variable is what they chose, but there's a function of the brand variable and you get paid off. 365 00:39:14.519 --> 00:39:19.019 According to this function, and this is how you can find. 366 00:39:19.019 --> 00:39:30.570 It happens a lot in lottery, whatever the thing right here. 367 00:39:30.570 --> 00:39:36.960 Okay. 368 00:39:36.960 --> 00:39:42.929 So. 369 00:39:44.159 --> 00:39:48.300 Another example is, um. 370 00:39:49.800 --> 00:39:54.059 I don't know, let's suppose you're buying an option on textbooks or something. 371 00:39:56.309 --> 00:40:01.139 Huh. 372 00:40:01.139 --> 00:40:04.800 Um. 373 00:40:06.389 --> 00:40:10.590 And maybe the strike price. 374 00:40:10.590 --> 00:40:15.300 Prices maybe make it a 1000, let's say. 375 00:40:15.300 --> 00:40:18.719 And so. 376 00:40:21.420 --> 00:40:25.800 Hello. 377 00:40:25.800 --> 00:40:28.920 Okay. 378 00:40:30.210 --> 00:40:34.920 Just clearly. 379 00:40:34.920 --> 00:40:39.179 Okay. 380 00:40:40.619 --> 00:40:44.010 On a close date. 381 00:40:45.210 --> 00:40:49.110 Okay, um, so you pay off. 382 00:40:51.329 --> 00:40:55.739 Excuse me. 383 00:41:03.119 --> 00:41:06.929 Hmm. 384 00:41:06.929 --> 00:41:10.679 Computers hungry. 385 00:41:12.030 --> 00:41:15.809 Huh. 386 00:41:15.809 --> 00:41:19.860 Hello. 387 00:41:23.940 --> 00:41:27.329 No worries. Okay. 388 00:41:27.329 --> 00:41:30.329 Hmm. 389 00:41:30.329 --> 00:41:36.179 You didn't show on the projectors that got a little 1. okay. Now. 390 00:41:37.800 --> 00:41:43.949 So, my random variable is what the test the thought closes that maybe when the office and expired. 391 00:41:43.949 --> 00:41:52.320 And I pay off, it takes less than a 1000. I get it. 392 00:41:52.320 --> 00:41:55.440 0, is. 393 00:41:55.440 --> 00:42:00.449 Greater than a 1000 I get paid off X minus the. 394 00:42:00.449 --> 00:42:03.690 Okay, okay now. 395 00:42:03.690 --> 00:42:08.699 So, how much money am I going to make? Um. 396 00:42:09.780 --> 00:42:13.800 You know, whatever, so, maybe you're going to say probability for 1100. 397 00:42:16.559 --> 00:42:21.480 Hello. 398 00:42:21.480 --> 00:42:27.630 Maybe, I don't know a quarter, probably 1200. 399 00:42:27.630 --> 00:42:33.059 The name and the very last night to a 1000. 400 00:42:33.059 --> 00:42:41.099 Hey, calls, I don't know, what is it, 58 or something like. 401 00:42:41.099 --> 00:42:48.539 So, what's the expected value that I'm assuming it can only be. 402 00:42:48.539 --> 00:42:52.889 Less than a 2100 Co founder. Okay. 403 00:42:54.750 --> 00:42:59.519 Goodbye. 404 00:43:02.039 --> 00:43:06.840 And let's say the sample space, it's a 900,000. 405 00:43:06.840 --> 00:43:12.269 11 11,200 dollars. 406 00:43:18.989 --> 00:43:24.599 Um, so what this thing works is, is I'm buying a. 407 00:43:24.599 --> 00:43:36.030 What the option means is 1000 is a Tesla is more than a 1000 dollars. What I win is by how much tesla's stock is over 1000. 408 00:43:36.030 --> 00:43:40.380 The tesla's stock is under a 1000. I did nothing. So this, um. 409 00:43:40.380 --> 00:43:43.530 Okay, and. 410 00:43:43.530 --> 00:43:46.710 So, the pay off is this is. 411 00:43:46.710 --> 00:43:50.610 Huh. 412 00:43:52.050 --> 00:43:57.179 I keep switching to another screen. There we go. Um. 413 00:44:02.130 --> 00:44:05.639 Huh. 414 00:44:07.650 --> 00:44:10.949 Okay, what I was trying to do was. 415 00:44:10.949 --> 00:44:15.329 Clean this up that is a less than. 416 00:44:21.389 --> 00:44:26.699 Well, I want to know how much I'm going to win on the average. Okay. 417 00:44:26.699 --> 00:44:36.239 So, the, my expected value that it's just the random variable. 418 00:44:36.239 --> 00:44:44.159 Well, the 5 days chance that the stock is no more than a 1000 phase off nothing. 419 00:44:44.159 --> 00:44:49.860 In 1 quarter chance that it's 1100 K through 100. 420 00:44:49.860 --> 00:44:53.519 But the 18 chance that it's 1200 phase 200. 421 00:44:53.519 --> 00:44:57.389 And the expected value of that was a. 422 00:44:57.389 --> 00:45:01.469 25 and 2550. 423 00:45:01.469 --> 00:45:05.250 So. 424 00:45:05.250 --> 00:45:09.480 Hello. 425 00:45:12.750 --> 00:45:18.539 Okay. 426 00:45:18.539 --> 00:45:25.739 So, they're finding expected value of a function of a random variable. Okay so this is again. 427 00:45:27.630 --> 00:45:37.110 Okay, sounds good. 428 00:45:37.110 --> 00:45:41.550 Okay, but the idea. 429 00:45:41.550 --> 00:45:47.849 Hello. 430 00:45:48.960 --> 00:45:52.079 That's a new idea for today from chapter 3. 431 00:45:52.079 --> 00:45:55.320 Another new idea so there's other, um. 432 00:45:56.789 --> 00:46:04.440 And this function is property expected values, the properties and so on. 433 00:46:04.440 --> 00:46:11.340 Um, you can add them in whatever. Um, so. 434 00:46:12.449 --> 00:46:18.389 I would do that, but let me give you another property. Um. 435 00:46:22.710 --> 00:46:28.829 Another property. 436 00:46:30.360 --> 00:46:35.130 So this is. 437 00:46:36.179 --> 00:46:39.510 Oh, great. 438 00:46:39.510 --> 00:46:46.440 The values are. 439 00:46:46.440 --> 00:46:51.480 Okay, so the various, the. 440 00:46:54.690 --> 00:46:57.780 They expect the value of X minus. 441 00:46:57.780 --> 00:47:01.980 Hello. 442 00:47:01.980 --> 00:47:05.880 Um. 443 00:47:05.880 --> 00:47:09.780 An example, um. 444 00:47:14.940 --> 00:47:18.449 Coin before. 445 00:47:20.610 --> 00:47:23.639 The probability of a 0 is 1 and a half. 446 00:47:23.639 --> 00:47:27.599 Okay. 447 00:47:29.190 --> 00:47:32.699 Expected value of X is the half. 448 00:47:32.699 --> 00:47:37.320 There are plus 4th. 449 00:47:37.320 --> 00:47:41.429 I just said that a few minutes ago. 450 00:47:44.400 --> 00:47:48.300 Sorry. 451 00:47:49.320 --> 00:47:52.860 Hmm. 452 00:47:58.320 --> 00:48:07.949 So, there's actually a number of people watching me on Webex. Okay. I got it. 453 00:48:09.210 --> 00:48:12.300 I got it. Yeah. 454 00:48:12.300 --> 00:48:15.989 Hello. 455 00:48:15.989 --> 00:48:19.920 Hello. 456 00:48:19.920 --> 00:48:25.230 Was that a question. 457 00:48:25.230 --> 00:48:29.429 So. 458 00:48:29.429 --> 00:48:37.619 The value X minus. 459 00:48:38.940 --> 00:48:42.420 You guys have. 460 00:48:42.420 --> 00:48:46.019 So. 461 00:48:48.389 --> 00:48:51.690 They're also either a 0 minus 1 half. 462 00:48:51.690 --> 00:48:55.530 1 would be 1 1 1. 463 00:48:56.849 --> 00:49:02.489 Sorry for that. 464 00:49:02.489 --> 00:49:07.349 Hello. 465 00:49:08.699 --> 00:49:13.230 So, um. 466 00:49:15.000 --> 00:49:23.849 So so what we have is. 467 00:49:24.989 --> 00:49:33.659 So probably probability on half. 468 00:49:35.099 --> 00:49:41.309 We have this thing here is X is 0 0 minus 1. 469 00:49:41.309 --> 00:49:44.400 Where do you order. 470 00:49:44.400 --> 00:49:50.099 Probably this 1 half 1 minus 1, half clarity with 1 quarter. 471 00:49:50.099 --> 00:49:54.059 So the very start that could be. 472 00:49:54.059 --> 00:50:00.780 Um, quarter, um. 473 00:50:00.780 --> 00:50:06.420 A quarter. 474 00:50:07.619 --> 00:50:11.130 Hello. 475 00:50:12.360 --> 00:50:17.699 And there's a new definition, the standard deviation. 476 00:50:18.989 --> 00:50:31.019 Wherever we went to the very end, and if we, if we draw the things here, these are values of X. and this is the probability. 477 00:50:31.019 --> 00:50:34.530 Mexico's 0, as a probability 1, half. 478 00:50:34.530 --> 00:50:38.670 Equals 1, the probability of 1 half. 479 00:50:38.670 --> 00:50:44.340 The expected the values in the middle and the standard deviation sort of gets the Fred. 480 00:50:44.340 --> 00:50:47.550 The other side of that. 481 00:50:47.550 --> 00:50:50.940 So the variant is larger. 482 00:50:50.940 --> 00:50:55.170 The numbers are more spread out smaller. 483 00:50:55.170 --> 00:51:04.619 Give me an example. So, the 1 point not to let me do a number of clients to binomial. Let's say. 484 00:51:04.619 --> 00:51:10.559 So this is the next new idea today is the function. 485 00:51:10.559 --> 00:51:14.489 The barriers that a random variable. 486 00:51:14.489 --> 00:51:18.900 And so the expected value the square of the different from. 487 00:51:21.449 --> 00:51:25.440 Okay, now. 488 00:51:27.659 --> 00:51:31.380 Another way to do the variance. 489 00:51:33.690 --> 00:51:37.170 I said it's expected value that Mike. 490 00:51:40.349 --> 00:51:45.389 So that we can spend it. 491 00:51:45.389 --> 00:51:51.929 Squared minus expected value of that. 492 00:51:51.929 --> 00:51:55.920 Expected value backs where. 493 00:51:55.920 --> 00:52:01.260 Okay, now there are some properties of expected value. 494 00:52:03.449 --> 00:52:11.670 I'll I'll hold it for a minute. 495 00:52:11.670 --> 00:52:15.119 Then I'll come back to that. 496 00:52:16.739 --> 00:52:22.739 Hello. 497 00:52:22.739 --> 00:52:25.949 Hello. 498 00:52:27.510 --> 00:52:33.630 That if I multiply the random variables, like some number, like C. 499 00:52:33.630 --> 00:52:40.769 I can pull it out. I can pull that out. 500 00:52:42.389 --> 00:52:45.630 Because I agree. 501 00:52:45.630 --> 00:52:48.780 Hello. 502 00:52:49.949 --> 00:52:54.630 And the reason is that. 503 00:52:54.630 --> 00:53:01.409 Did you check the values integral. 504 00:53:04.949 --> 00:53:08.820 The probability of each value of that. 505 00:53:08.820 --> 00:53:17.159 The times that the probability of that value is that. 506 00:53:17.159 --> 00:53:25.139 Both the. 507 00:53:25.139 --> 00:53:29.639 So, I can pull factors out of the expectations. So. 508 00:53:29.639 --> 00:53:38.940 Another property is that if I have to random variables. 509 00:53:39.989 --> 00:53:43.920 The thumb that that will. 510 00:53:43.920 --> 00:53:50.400 Good so. 511 00:53:52.230 --> 00:53:59.340 Let me give you an example of. 512 00:53:59.340 --> 00:54:04.860 Well, 1st prove that because that's the sum of all the. 513 00:54:08.429 --> 00:54:15.360 They'll give that particular value of. 514 00:54:15.360 --> 00:54:19.469 I'll I'll wait, I'll do this. 515 00:54:19.469 --> 00:54:26.130 I mentioned this on Thursday actually, but you can split up that the fact that value. 516 00:54:26.130 --> 00:54:29.730 But this is also, this is true regardless. 517 00:54:33.690 --> 00:54:45.179 Any correlation that's always. 518 00:54:45.179 --> 00:54:51.389 Hello. 519 00:54:52.769 --> 00:54:58.769 So, I expected value that some of 2 terms, whatever they are, I can break it up. 520 00:55:00.420 --> 00:55:05.159 So. 521 00:55:09.630 --> 00:55:12.750 Hey. 522 00:55:12.750 --> 00:55:17.820 So, what this means is now for the barrier starting location. 523 00:55:17.820 --> 00:55:21.630 Virtual. 524 00:55:21.630 --> 00:55:30.329 So, the various I said is expected value of X minus the suspect of value. 525 00:55:30.329 --> 00:55:37.260 Square, so the square means it's emphasizing big barrier. 526 00:55:37.260 --> 00:55:43.980 Various off the, the little there so that was the expected value of expert. 527 00:55:43.980 --> 00:55:47.849 To the value that. 528 00:55:47.849 --> 00:55:52.139 Expected value where. 529 00:55:53.969 --> 00:55:57.300 Close the expected value that is cleared. 530 00:55:57.300 --> 00:56:02.579 Inside the thing in February back to the cost so this would be. 531 00:56:02.579 --> 00:56:07.380 To expect the value back time. This is that. 532 00:56:08.400 --> 00:56:11.400 And the square foot that. 533 00:56:12.449 --> 00:56:18.570 Where my, um. 534 00:56:23.730 --> 00:56:28.860 So this is another way to do Barry. Let me write that down. Okay. 535 00:56:39.599 --> 00:56:44.969 We're going to find it. 536 00:56:44.969 --> 00:56:49.800 And, um. 537 00:56:51.269 --> 00:56:56.460 The only thing I'm doing this on real computers, this may give. 538 00:56:56.460 --> 00:57:00.449 Numerical round off errors because. 539 00:57:00.449 --> 00:57:06.960 These 2 terms may be very large and be very close to each other. 540 00:57:06.960 --> 00:57:10.199 So then when you subtract them, you may lose significant. 541 00:57:10.199 --> 00:57:13.530 That's all good. Good people worried about significant. 542 00:57:13.530 --> 00:57:19.199 On, which is kind of give an example. 543 00:57:22.409 --> 00:57:25.679 Hello. 544 00:57:25.679 --> 00:57:29.610 There was 1 time. 545 00:57:29.610 --> 00:57:34.050 Value that's equals 1 half. 546 00:57:34.050 --> 00:57:37.349 Expected value that squared. 547 00:57:37.349 --> 00:57:43.019 Well, it's also what happened to save is that 1. 548 00:57:43.019 --> 00:57:48.059 So, the variance of that is 1 half. 549 00:57:48.059 --> 00:57:54.329 1 quarter equals 1 quarter I need to do it. 550 00:57:54.329 --> 00:57:59.670 Okay. 551 00:58:01.469 --> 00:58:07.110 For. 552 00:58:09.360 --> 00:58:14.460 Okay, here. 553 00:58:14.460 --> 00:58:17.789 Yeah. 554 00:58:19.199 --> 00:58:23.849 It was another comment. 555 00:58:23.849 --> 00:58:27.119 Hello. 556 00:58:28.559 --> 00:58:38.489 So this is the case where computers are worse than the old fashioned blackboard the old classrooms with thought boards around all the walls. 557 00:58:38.489 --> 00:58:42.960 Let's go, you can go more towards the board, but. 558 00:58:42.960 --> 00:58:47.250 Okay, but I couldn't stand the PDF. 559 00:58:48.900 --> 00:58:53.550 Oh, okay. Um. 560 00:58:56.940 --> 00:59:00.389 Let me give another example. 561 00:59:02.190 --> 00:59:06.179 Hello. 562 00:59:07.619 --> 00:59:12.059 I know me oh, here. 563 00:59:13.170 --> 00:59:20.219 The coin 4 times the close, the number, and then. 564 00:59:22.349 --> 00:59:25.829 For her does not matter. 565 00:59:27.300 --> 00:59:32.369 Um. 566 00:59:32.369 --> 00:59:40.409 So, the probability of a particular number of K peg or Tuesday. 567 00:59:40.409 --> 00:59:45.449 Uh, SharePoint. 568 00:59:45.449 --> 00:59:49.380 Or Tuesday, times of. 569 00:59:50.909 --> 00:59:54.119 Hello. 570 00:59:55.170 --> 00:59:58.590 After the day. 571 00:59:58.590 --> 01:00:02.099 That was 1 half to the forum line yesterday. 572 01:00:02.099 --> 01:00:08.070 For Tuesday, 16. 573 01:00:08.070 --> 01:00:13.530 So, the probabilities are. 574 01:00:16.260 --> 01:00:21.570 16 right right. 575 01:00:25.619 --> 01:00:29.489 0, is, is 116 probably 1 that. 576 01:00:29.489 --> 01:00:36.059 Is 415 260. 577 01:00:36.059 --> 01:00:39.210 The 3 guys, the 408. 578 01:00:39.210 --> 01:00:45.989 Forehead is 100 expected value. 579 01:00:47.489 --> 01:00:51.000 0 times 116 plus 1 time. 580 01:00:51.000 --> 01:00:56.159 462 5 616. 581 01:00:56.159 --> 01:01:01.769 3 times 415 plus 1 before. 582 01:01:01.769 --> 01:01:12.960 116 equals 0 plus 4 plus 12, which makes 16 plus 12, which makes 28 look forward to make 32. 583 01:01:12.960 --> 01:01:16.949 Okay. 584 01:01:16.949 --> 01:01:23.880 The prize I cost a coin 4 times they expect the number of factors to. 585 01:01:23.880 --> 01:01:30.059 Okay, the various. 586 01:01:30.059 --> 01:01:38.820 But what's the expected number of X squared 0 times 116 plus 1 time 416 plus 4. 587 01:01:38.820 --> 01:01:43.980 615 plus 9 times 415. 588 01:01:43.980 --> 01:01:47.250 But 16 at 150. 589 01:01:48.360 --> 01:01:55.500 04, plus 24, which makes 28 plus 36 and 28. 590 01:01:55.500 --> 01:01:59.039 Uh, let's. 591 01:01:59.039 --> 01:02:03.420 50 or something, but 16, which may. 592 01:02:03.420 --> 01:02:09.150 Are you doing. 593 01:02:09.150 --> 01:02:16.980 Definitely on that. Not completely convinced the baby is correct correct? 594 01:02:16.980 --> 01:02:20.130 Sorry. 595 01:02:20.130 --> 01:02:24.030 70 on, let's say 4. 596 01:02:24.030 --> 01:02:30.480 Go back with 16, but 36th that is 52. 597 01:02:30.480 --> 01:02:38.309 Plus, 24, which may 76 plus 4, 8. 598 01:02:38.309 --> 01:02:42.869 Okay. 599 01:02:42.869 --> 01:02:46.079 Okay. 600 01:02:46.079 --> 01:02:52.110 The only 2 people in the room that can do this stuff. 601 01:02:52.110 --> 01:02:55.679 Yeah, okay. Um. 602 01:02:55.679 --> 01:03:01.230 And so the variance. 603 01:03:01.230 --> 01:03:07.500 Expected number of X squared minus, expect the number of X. 604 01:03:08.639 --> 01:03:14.760 Where do you quote? 5 minus 4 equals 1. 605 01:03:14.760 --> 01:03:18.239 And it's not going to be 1, so. 606 01:03:19.800 --> 01:03:24.210 Um, so okay. 607 01:03:25.530 --> 01:03:30.840 Hello. 608 01:03:30.840 --> 01:03:39.780 If I bought it, I could, I could swap the what's called the probability match. So. 609 01:03:41.280 --> 01:03:48.030 Huh. 610 01:03:48.030 --> 01:03:51.150 A loyalty. 611 01:03:51.150 --> 01:03:55.349 And we're going to get 1. 612 01:03:55.349 --> 01:03:58.980 For so this would. 613 01:04:00.059 --> 01:04:05.969 The 116 1 is going to be. 614 01:04:05.969 --> 01:04:10.050 416 to. 615 01:04:10.050 --> 01:04:13.380 3, 3, 3, 4. 616 01:04:13.380 --> 01:04:16.829 For from the 1. okay. 617 01:04:16.829 --> 01:04:23.519 And, um, and so the mean is going to be here. 618 01:04:23.519 --> 01:04:30.269 And for the variances 1, so the variance will be, but. 619 01:04:30.269 --> 01:04:34.860 So that's fair. That's fair. Yeah. 620 01:04:34.860 --> 01:04:43.170 Um, so I have another new idea here of finding the probability that function. 621 01:04:43.170 --> 01:04:46.800 Hello. 622 01:04:46.800 --> 01:04:52.230 Hello. 623 01:04:58.349 --> 01:05:02.969 Um, let me do another 1 on this, um. 624 01:05:04.019 --> 01:05:09.750 What are the 5 a variable number of coin? 625 01:05:09.750 --> 01:05:14.010 And. 626 01:05:14.010 --> 01:05:21.150 No, not just for. Okay, so, um. 627 01:05:22.320 --> 01:05:26.670 The problem is okay. 628 01:05:26.670 --> 01:05:30.570 And Tuesday, and I've got to be fair coins. Okay. 629 01:05:33.750 --> 01:05:38.909 Other unfair points later issues. Okay. 630 01:05:38.909 --> 01:05:44.519 Expected value of. 631 01:05:44.519 --> 01:05:50.280 At the top of all the well, the value, which is hey. 632 01:05:50.280 --> 01:05:54.329 And then Tuesday. 633 01:05:54.329 --> 01:05:59.159 And I did that last time, I think. Oh. 634 01:05:59.159 --> 01:06:05.280 This will, um. 635 01:06:05.280 --> 01:06:13.500 Yeah, I get something slightly different than me review with some of the issues. Hey, this, this level of all the. 636 01:06:13.500 --> 01:06:19.590 A little bit. 637 01:06:19.590 --> 01:06:25.050 They are here, so. 638 01:06:28.829 --> 01:06:35.280 And I could pull it in now and this was that out to. 639 01:06:36.539 --> 01:06:41.429 And the sum of our factorial over. 640 01:06:43.980 --> 01:06:48.090 I'm going to say. 641 01:06:48.090 --> 01:06:52.889 Yeah, right. 642 01:06:55.650 --> 01:07:01.199 Um, because it's a 2 K. 643 01:07:01.199 --> 01:07:05.820 Probability of the day of probability to the ad right today. 644 01:07:05.820 --> 01:07:09.420 And both of those, and the 1st few reports 1 app. 645 01:07:11.940 --> 01:07:17.309 Okay, that's the good. 646 01:07:20.070 --> 01:07:24.599 The practices around it. 647 01:07:24.599 --> 01:07:34.110 But, in case, if I wave my hands a little bit, cause it's getting late. Um. 648 01:07:36.269 --> 01:07:39.989 The 2nd value is actually going to be at over to. 649 01:07:41.610 --> 01:07:48.510 I can hit it in more detail and then the next question is, what's the barrier. 650 01:07:50.550 --> 01:07:54.059 And that's going to be the sum of all the things where. 651 01:07:54.059 --> 01:07:59.369 Time and to to live with that. 652 01:08:01.440 --> 01:08:04.619 Hello? Hello? Hello? 653 01:08:04.619 --> 01:08:11.400 We may move that to Thursday. 654 01:08:11.400 --> 01:08:18.180 So, the new ideas, just to review, we did some examples with faith there. 655 01:08:19.979 --> 01:08:25.020 Which is conditional probability backward. 656 01:08:31.649 --> 01:08:37.020 Hey. 657 01:08:37.020 --> 01:08:40.350 And we saw function of a. 658 01:08:40.350 --> 01:08:43.380 Functions over at a variable. 659 01:08:43.380 --> 01:08:47.340 And we saw more on what we expected values. 660 01:08:47.340 --> 01:08:51.960 We felt very so that. 661 01:08:51.960 --> 01:08:57.869 And what I'll do next time is these on more granular distributions so. 662 01:08:57.869 --> 01:09:04.079 I mean, I may drag in a touch chapter. 663 01:09:04.079 --> 01:09:09.149 2, I'd like to make things up a little more just. 664 01:09:09.149 --> 01:09:14.670 I'm reading it. 665 01:09:15.960 --> 01:09:22.350 And then a question on the floor. 666 01:09:22.350 --> 01:09:25.739 I can. 667 01:09:25.739 --> 01:09:29.399 Hello. 668 01:09:29.399 --> 01:09:32.460 Alright, do me a little bit. 669 01:09:34.859 --> 01:09:42.449 Which do you prefer for the midterm exam? The 1st exam 2428. 670 01:09:42.449 --> 01:09:47.369 If you put on the classes. 671 01:09:47.369 --> 01:09:53.699 Okay, I'm here are 28, 8, 8, 8. 672 01:09:53.699 --> 01:10:02.130 The 2828, I'm hearing 8 they say. 673 01:10:02.130 --> 01:10:05.489 It cannot be right I mean. 674 01:10:05.489 --> 01:10:10.020 28th. 675 01:10:10.020 --> 01:10:14.609 Okay, in class in page, whatever. 676 01:10:14.609 --> 01:10:18.270 And if you're going to be, uh. 677 01:10:18.270 --> 01:10:24.689 I don't know what we do. We trust you basically have to see what else can I do. 678 01:10:24.689 --> 01:10:28.050 So, I'll sit here if you have questions. 679 01:10:28.050 --> 01:10:32.399 Also, remember to me, write that down. 680 01:10:32.399 --> 01:10:35.970 So. 681 01:10:40.020 --> 01:10:45.720 Hello. 682 01:10:45.720 --> 01:10:51.119 You're actually very 1st of all with the. 683 01:10:51.119 --> 01:10:55.380 Hello. 684 01:10:55.380 --> 01:11:03.119 What's your thoughts on. 685 01:11:03.119 --> 01:11:16.920 Yeah, okay. The universe. 686 01:11:16.920 --> 01:11:21.239 Here. 687 01:11:21.239 --> 01:11:32.640 That's the universe now. 688 01:11:32.640 --> 01:11:40.829 I don't know. 689 01:11:40.829 --> 01:11:50.430 I would like to think about that. Right? So, engineers have given me the. 690 01:11:50.430 --> 01:11:54.270 76. 691 01:11:54.270 --> 01:11:59.609 I think that's it. 692 01:11:59.609 --> 01:12:03.600 For undergrad given engineer left for. 693 01:12:03.600 --> 01:12:06.600 I think. 694 01:12:07.619 --> 01:12:16.829 Yeah, the last part of the engineers for engineers. 695 01:12:16.829 --> 01:12:20.220 Like, if if it's given under grad. 696 01:12:20.220 --> 01:12:23.279 Can you look inside that right here? 697 01:12:23.279 --> 01:12:28.710 If it's undergrad given engineers, you look inside so. 698 01:12:28.710 --> 01:12:34.500 Sure, sure this 1 was 3000 to 1000. 699 01:12:34.500 --> 01:12:38.729 I'm not sure. Oh, really? 700 01:12:38.729 --> 01:12:43.560 No, I thought it would be fine. 701 01:12:43.560 --> 01:12:47.430 I really I. 702 01:12:47.430 --> 01:12:51.449 It's. 703 01:12:51.449 --> 01:12:54.810 Okay. 704 01:12:54.810 --> 01:13:03.090 Uh. 705 01:13:03.090 --> 01:13:06.390 The normal. 706 01:13:06.390 --> 01:13:10.590 Hello. 707 01:13:10.590 --> 01:13:16.470 Huh. 708 01:13:16.470 --> 01:13:20.250 Bye. 709 01:13:20.250 --> 01:13:24.449 This is the. 710 01:13:24.449 --> 01:13:29.100 Right, right? 711 01:13:29.100 --> 01:13:35.729 And then you. 712 01:13:35.729 --> 01:13:39.390 Do it that will work. 713 01:13:39.390 --> 01:13:46.920 Well, we're well examples, not at all, but okay I can do that. Yeah. Yeah. 714 01:13:46.920 --> 01:13:49.979 Perfect. 715 01:13:49.979 --> 01:13:53.369 For that. 716 01:13:53.369 --> 01:14:02.850 Okay, so we do and you just get an undergrad switch, hundreds of rejection. 717 01:14:02.850 --> 01:14:06.689 Undergrad at the end date, but. 718 01:14:08.250 --> 01:14:13.319 It always be. 719 01:14:13.319 --> 01:14:21.000 Wouldn't be for. 720 01:14:21.000 --> 01:14:24.090 All that kind of things. 721 01:14:24.090 --> 01:14:27.630 You're right. 722 01:14:27.630 --> 01:14:34.500 And so on and set up. 723 01:14:34.500 --> 01:14:38.729 So. 724 01:14:38.729 --> 01:14:42.600 Is it possible to instead of doing this 1? 725 01:14:42.600 --> 01:14:50.010 That's just the 1 seminar your work your way anyway in here is to give him some of the rest of it. 726 01:14:51.630 --> 01:14:55.109 113. 727 01:14:55.109 --> 01:14:58.289 Oh. 728 01:14:58.289 --> 01:15:02.100 Wow. 729 01:15:02.100 --> 01:15:06.510 1013. 730 01:15:06.510 --> 01:15:09.930 Yeah, lunch them to do, right? 731 01:15:09.930 --> 01:15:14.069 I don't know what we have for example, but, um. 732 01:15:14.069 --> 01:15:17.430 Well, I did, for example. 733 01:15:17.430 --> 01:15:23.039 I found the probability that you're aware will get the positive. 734 01:15:23.039 --> 01:15:31.710 We don't have probably a team. 735 01:15:31.710 --> 01:15:36.600 Right, but you can find with the total profitability there because. 736 01:15:37.619 --> 01:15:42.090 You've got the, and probably the key is, if not the. 737 01:15:44.069 --> 01:15:52.529 Yeah, you have to calculate that. That's the denominator of the page. Yeah. Can you tell me. 738 01:15:52.529 --> 01:15:56.729 So, I have this 1, so I know that's. 739 01:15:56.729 --> 01:16:02.460 Yeah. 740 01:16:02.460 --> 01:16:08.159 You can work on this thing here like, I guess. 741 01:16:08.159 --> 01:16:11.220 That's what I. 742 01:16:11.220 --> 01:16:14.729 For the. 743 01:16:14.729 --> 01:16:20.909 And this can be made. 744 01:16:20.909 --> 01:16:24.270 Hello. 745 01:16:24.270 --> 01:16:34.260 Probability that do not guarantee is a probability and the. 746 01:16:34.260 --> 01:16:39.390 Divided by the you gotta get the key so. 747 01:16:39.390 --> 01:16:45.930 The thing, so you have to probably T, not and I also need to probably add them to the table. 748 01:16:45.930 --> 01:16:51.180 The problem. 749 01:16:51.180 --> 01:16:55.229 See. 750 01:16:56.369 --> 01:17:01.439 Good. 751 01:17:01.439 --> 01:17:14.729 The problem the problem with the. 752 01:17:14.729 --> 01:17:19.020 Can you come back so. 753 01:17:19.020 --> 01:17:23.399 Thank you. 754 01:17:26.069 --> 01:17:29.939 Okay, Mm hmm. 755 01:17:29.939 --> 01:17:35.699 You got the 1st, part of the, the 2nd part we got to do something like that also. 756 01:17:35.699 --> 01:17:39.930 The problem of the. 757 01:17:42.989 --> 01:17:49.920 Can you go on and Bill and given the fact that. 758 01:17:49.920 --> 01:17:53.460 You know, I read something like that. 759 01:17:53.460 --> 01:17:57.510 I'm just not sure it is. And. 760 01:17:57.510 --> 01:18:04.619 It's more of a problem. Yeah, the probability of key is probably the key. 761 01:18:04.619 --> 01:18:11.369 And the not pass, the probability is the portfolio probability that. 762 01:18:11.369 --> 01:18:20.279 Hello, so I think. 763 01:18:22.170 --> 01:18:25.260 You know. 764 01:18:25.260 --> 01:18:28.260 And I'll give you some time, and I probably actually put it on. 765 01:18:28.260 --> 01:18:35.819 Yeah, I have the same question, but the explanation not given to the AP. 766 01:18:35.819 --> 01:18:43.439 I need to remind you, I'll give you. 767 01:18:43.439 --> 01:18:48.899 Weekend you have the. 768 01:18:48.899 --> 01:18:53.640 And I'm also curious if you keep any of the factory. 769 01:18:53.640 --> 01:18:57.810 Oh, yeah, we can put forth. 770 01:18:57.810 --> 01:19:00.960 Off the hook. 771 01:19:00.960 --> 01:19:03.989 Awesome. 772 01:19:03.989 --> 01:19:08.550 It's, it's there, um. 773 01:19:08.550 --> 01:19:12.779 Hello. 774 01:19:12.779 --> 01:19:17.159 I was going to. 775 01:19:17.159 --> 01:19:27.689 I'm trying to get it up. Well, yeah, I mean, there are people that grab the exam, but you can also have a direction. 776 01:19:27.689 --> 01:19:35.939 Okay, Mike. Thanks, Eva. And that's 31 spring 2021 facility at all. 777 01:19:35.939 --> 01:19:40.800 Hello. 778 01:19:40.800 --> 01:19:53.609 Oh, teaching okay. I have to check where it is. Exactly. Oh, it's not very. 779 01:19:53.609 --> 01:20:02.159 Oh, God, pull it back. If you go back more than 1. 780 01:20:02.159 --> 01:20:05.850 Testing. 781 01:20:05.850 --> 01:20:12.869 1, with solutions. 782 01:20:12.869 --> 01:20:27.210 Cool, and what I do is I recycle about a 3rd of the numbers so if you study all the old exam field over the course, thank you, you'll work hard. 783 01:20:27.210 --> 01:20:33.390 Yeah, but what we're trying to do is migrate exams to great scope. 784 01:20:33.390 --> 01:20:38.579 Because with remote people would also be creating easier. 785 01:20:38.579 --> 01:20:43.050 So. 786 01:20:43.050 --> 01:20:47.039 Hello. 787 01:20:48.539 --> 01:20:53.970 Okay. 788 01:20:53.970 --> 01:20:56.970 So. 789 01:21:00.090 --> 01:21:03.569 No. 790 01:21:06.960 --> 01:21:11.489 That became more now. 791 01:21:11.489 --> 01:21:18.149 Yeah, I have a good day. 792 01:21:18.149 --> 01:21:23.609 Yeah. 793 01:21:31.020 --> 01:21:36.119 Okay. 794 01:22:58.289 --> 01:23:04.949 Hello. 795 01:23:07.380 --> 01:23:12.420 Huh. 796 01:23:12.420 --> 01:23:16.920 Hello.