WEBVTT 1 00:00:00.000 --> 00:00:16.649 Silence. 2 00:00:22.260 --> 00:00:34.770 Silence. 3 00:00:34.770 --> 00:00:39.359 Silence. 4 00:00:47.280 --> 00:00:51.479 Silence. 5 00:00:57.960 --> 00:01:07.469 Oh, okay, good afternoon class. My universal question is. 6 00:01:07.469 --> 00:01:15.420 A, 2nd, here. 7 00:01:26.670 --> 00:01:30.030 Silence. 8 00:01:30.030 --> 00:01:34.829 Okay, 1. 9 00:01:34.829 --> 00:01:45.269 Anyone Thank you. Okay. Hope you had a good weekend. So this is engineering probability classify. 10 00:01:45.269 --> 00:01:55.079 That Bates 2021 and we will see how the hardware works today. 11 00:01:55.079 --> 00:02:01.650 You are free, just being a probability class to make the back on. 12 00:02:01.650 --> 00:02:07.950 Which hardware, which particular piece of hardware will fail? 1st today. 13 00:02:07.950 --> 00:02:14.580 I will be asking for my usual 10%. You can keep the commission. 14 00:02:14.580 --> 00:02:19.889 I'm going to good. Okay, so this is, um. 15 00:02:21.569 --> 00:02:27.300 And it's class 5 and. 16 00:02:27.300 --> 00:02:35.039 Look okay, so what are we going to do today? We're continuing on with. 17 00:02:36.419 --> 00:02:42.000 Probability stuff and so on and stuff like that. 18 00:02:42.000 --> 00:02:47.490 For the moment, and by the way what this show am I using. 19 00:02:47.490 --> 00:02:51.569 X. 20 00:02:51.569 --> 00:02:56.310 Okay, so. 21 00:02:58.590 --> 00:03:07.020 If you'd like to see, I'll show you about my machine if you want to get a free thing. 22 00:03:12.300 --> 00:03:17.520 Next 5.8. 23 00:03:19.050 --> 00:03:23.939 Silence. 24 00:03:23.939 --> 00:03:29.400 And if you want to see what my machine is, like. 25 00:03:29.400 --> 00:03:41.250 It has a 128 gigabytes of DRAM and it's a dual 6 core Z on with error correcting memory. 26 00:03:42.840 --> 00:03:48.360 Okay, so. 27 00:03:51.270 --> 00:03:56.340 Assuming that you can, I'm going to try to log in with a 2nd window just. 28 00:03:56.340 --> 00:04:01.110 Or maybe I'm not that tends to screw things up. Okay so. 29 00:04:02.580 --> 00:04:06.479 The theory is that you are, um. 30 00:04:07.650 --> 00:04:12.659 Seeing 1 of the Windows is the blog for the course. 31 00:04:12.659 --> 00:04:23.910 I don't actually dare try to log myself in a 2nd time, because I'm thinking this is possibly some of the problems. So. 32 00:04:25.829 --> 00:04:32.158 So, if you're seeing the screen, actually. 33 00:04:36.629 --> 00:04:41.218 The theory is that you're seeing the screen if you are not. 34 00:04:42.538 --> 00:04:46.858 Then then, tell me. 35 00:04:46.858 --> 00:04:50.968 But here's the funny thing about my laptop that. 36 00:04:52.014 --> 00:05:06.324 My 6 year old laptop tends to be more reliable for the course than this new laptop that I just bought in to go figure. Okay we have 2. excellent. So, Shawn and Meg, and those are our CS IDs. 37 00:05:06.324 --> 00:05:16.884 So, their email addresses are the, they've got office hours that they are going to be running on Webex meet. 38 00:05:17.423 --> 00:05:30.353 There and to the idea is that if not to appears towards us, started the office hour, then they are, then they're going to go away. On the other hand. 39 00:05:30.353 --> 00:05:33.894 They want to help you, they acknowledge you're paying money for the course. 40 00:05:35.038 --> 00:05:44.848 Good money and therefore they want to help you succeed and so feel free to contact them for help at other times and they'll try to be available. 41 00:05:44.848 --> 00:05:56.848 By the way you guys may not realize it, but our Pi gets brownie points from the percentage of students who actually graduate and this is 1 of the ways we're better than. 42 00:05:56.848 --> 00:06:02.278 Some of the less colleges, so we have a totally selfish reason. And you're actually graduating. 43 00:06:02.278 --> 00:06:07.649 Apart from the fact that reflected glory of the cross feel good when our students succeed. 44 00:06:07.649 --> 00:06:15.059 So, even if they don't buy us new art centers okay. Now, probability. 45 00:06:15.059 --> 00:06:21.899 I just moved this over from class for blog, because this is stuff that we haven't seen. 46 00:06:23.278 --> 00:06:28.738 I haven't seen so much, but already a capture this. 47 00:06:28.738 --> 00:06:33.149 Down down actually. 48 00:06:37.499 --> 00:06:41.249 Okay, that's better. 49 00:06:41.249 --> 00:06:45.298 Okay. 50 00:06:45.298 --> 00:06:50.759 So, we've got sampling. 51 00:06:50.759 --> 00:06:54.569 With without. 52 00:06:54.569 --> 00:06:59.608 Replacement. 53 00:06:59.608 --> 00:07:04.019 And with or without ordering. 54 00:07:04.019 --> 00:07:09.119 Okay, so. 55 00:07:10.528 --> 00:07:15.418 What I mean, is that suppose. 56 00:07:15.418 --> 00:07:21.899 Let me, I don't know why don't we go back here? 57 00:07:21.899 --> 00:07:26.879 2 quarter and dime. 58 00:07:26.879 --> 00:07:30.209 Okay, so that is my. 59 00:07:30.209 --> 00:07:33.658 Um. 60 00:07:33.658 --> 00:07:38.369 Silence. 61 00:07:38.369 --> 00:07:41.488 And it was 2 coins. 62 00:07:41.488 --> 00:07:46.048 And I'm going to draw. 63 00:07:46.048 --> 00:07:49.468 2 coins. 64 00:07:49.468 --> 00:07:54.689 Now, so it says. 65 00:07:54.689 --> 00:07:58.439 Was replacement. 66 00:07:58.439 --> 00:08:01.978 Then it could be. 67 00:08:01.978 --> 00:08:05.759 2. D. D. D. they in quarter. 68 00:08:05.759 --> 00:08:16.978 Quiz quarter, because time replacement is replacement with order. Okay. 69 00:08:18.449 --> 00:08:22.499 Replacement without. 70 00:08:22.499 --> 00:08:26.819 Then it's on the on basically. 71 00:08:26.819 --> 00:08:31.889 The set of Q, and date is there's 2 cases here this 1 case. 72 00:08:31.889 --> 00:08:36.629 With replacement. 73 00:08:36.629 --> 00:08:42.599 Then we could get to to date. 74 00:08:42.599 --> 00:08:46.918 Day to day. 75 00:08:47.938 --> 00:08:56.668 4 cases yes. So I think, uh, people can see what you're writing. So are they asking, like. 76 00:08:56.668 --> 00:09:01.739 In lives the campaign, the chat I can't see. 77 00:09:03.538 --> 00:09:06.629 I could full screen at something like this. 78 00:09:06.629 --> 00:09:10.769 Is it that you can't see the window at all or? It's too small. 79 00:09:11.788 --> 00:09:16.649 Isn't it just too small? And now it's good. Okay you maximize. I think we're okay. 80 00:09:16.649 --> 00:09:30.418 Yeah, okay. I need a better replacement for the overcome. Okay so I'm doing 2 coins and replacement and with ordering, it could be. 81 00:09:30.418 --> 00:09:34.739 Somebody needs to mute yeah. To get killed. 82 00:09:34.739 --> 00:09:42.958 And 1, and I'm very excited. 83 00:09:42.958 --> 00:09:46.499 So somebody needs to get. 84 00:09:46.499 --> 00:09:53.908 You know, tag each other. Oops. Okay. 85 00:09:53.908 --> 00:09:57.749 I'm back to here. Okay. I'm. 86 00:09:57.749 --> 00:10:03.389 With replacement with. Okay, so, 2 cases without replacement with ordering. 87 00:10:03.389 --> 00:10:10.889 Quarter dime down quarter 1 case without replacement with, without ordering just a set of quarter guide. 88 00:10:10.889 --> 00:10:19.828 4 cases with replacement with adding because the 1st client could be quarter a dime the 2nd climb quarter a dime and the final thing would be. 89 00:10:19.828 --> 00:10:25.198 Um, with replacement and. 90 00:10:25.198 --> 00:10:28.948 Without. 91 00:10:28.948 --> 00:10:36.028 And that would be so set notation without order I could be 2 quarters quarter in a dime or. 92 00:10:36.028 --> 00:10:40.948 2 dimes and 3 cases, so, 4 different ways you can look at that. 93 00:10:40.948 --> 00:10:44.548 And each of them has at different times. 94 00:10:44.548 --> 00:10:48.239 I use now. 95 00:10:48.239 --> 00:10:53.038 So, how do we get the numbers here? 96 00:10:54.208 --> 00:10:59.548 Well, whatever replacement with oh, I'm assuming the coins are different. I've only got 1 quote. Okay. 97 00:10:59.548 --> 00:11:04.528 Well, here is basically and raise. 98 00:11:04.528 --> 00:11:11.788 To pick the 1st coin, and then N minus 1 and so on. So the number of ways here. 99 00:11:11.788 --> 00:11:16.349 Is in fact, trial minus K plus 1 factorial. 100 00:11:16.349 --> 00:11:19.769 That's regardless is cake was too. 101 00:11:19.769 --> 00:11:22.798 Without replacing without order. 102 00:11:22.798 --> 00:11:26.249 Then then that would be end choose K. 103 00:11:26.249 --> 00:11:29.278 Which would be N, factor or K factor. 104 00:11:29.278 --> 00:11:33.658 In minus K factor replacement with order. 105 00:11:33.658 --> 00:11:38.399 It would be end to the K, because we each coin. 106 00:11:38.399 --> 00:11:44.068 Right each call, we're going to end choices for a place without order. 107 00:11:44.068 --> 00:11:49.168 I'll think about that. 108 00:11:49.168 --> 00:11:52.979 So it's a note. 109 00:11:52.979 --> 00:11:57.688 Yeah, I'll come back to that. 1. okay, so. 110 00:12:01.198 --> 00:12:06.658 Actually, let me see here. 111 00:12:06.658 --> 00:12:11.969 Silence. 112 00:12:13.499 --> 00:12:19.619 Okay, I'll make it bigger when I go writing in it. 113 00:12:19.619 --> 00:12:23.908 Okay, so these are just different possibilities here. 114 00:12:23.908 --> 00:12:28.109 And I got some examples of some of them on homework and so on. 115 00:12:31.678 --> 00:12:37.589 And I just ran over them. 116 00:12:37.589 --> 00:12:43.438 Let me catch up, I want to go back here and make certain that I have the. 117 00:12:44.578 --> 00:12:48.658 Chat window off. Oh, okay. Good. 118 00:12:52.703 --> 00:13:06.864 Okay, 2 bars, 2 from 5. so I base it a nice example of that. Okay. Now we get cases like, we drive profitability that they're all different. 119 00:13:06.864 --> 00:13:07.764 Let's say. 120 00:13:08.068 --> 00:13:12.688 So that. 121 00:13:17.009 --> 00:13:22.499 Well, I'll do exactly the example that I have. 122 00:13:22.499 --> 00:13:26.158 Silence. 123 00:13:26.158 --> 00:13:30.178 Silence. 124 00:13:38.969 --> 00:13:45.178 Okay, so if I'm drawing. 125 00:13:48.149 --> 00:14:02.099 There we go, so, 3 balls from 5 width repipe them. They're all different. 126 00:14:02.099 --> 00:14:09.058 Well, number cases where they're all different than. 127 00:14:10.078 --> 00:14:14.788 Again, so so this is point Tom, like, 8 here. 128 00:14:14.788 --> 00:14:21.509 With replacement. 129 00:14:21.509 --> 00:14:25.168 Well, then the. 130 00:14:25.168 --> 00:14:28.708 1st, 5 choices. 131 00:14:29.908 --> 00:14:35.278 2nd 535 said 125 choices. 132 00:14:35.278 --> 00:14:39.869 Now. 133 00:14:39.869 --> 00:14:42.899 Number of cases where they're different. 134 00:14:42.899 --> 00:14:49.739 Silence. 135 00:14:49.739 --> 00:14:53.879 It's 5 times, 4 times, 3 equal 6 day. 136 00:14:55.558 --> 00:14:59.489 So, the probability that they're all different. 137 00:14:59.489 --> 00:15:03.239 It will be 61 and 25. 138 00:15:03.239 --> 00:15:17.849 Okay, permutations and drawing and objects. That's a number of ways. We can order the objects of permutations. 139 00:15:20.668 --> 00:15:26.879 Number of ways. 140 00:15:26.879 --> 00:15:30.149 And objects. 141 00:15:30.149 --> 00:15:35.969 So, and that's called your factorial. 142 00:15:35.969 --> 00:15:39.778 So, for example. 143 00:15:39.778 --> 00:15:43.288 Factor 6, if I got 3 objects. 144 00:15:43.288 --> 00:15:47.399 A. B. C. A. B. C. AC. 145 00:15:47.399 --> 00:15:53.369 B, a. C. A. C. A. C. A. B. 146 00:15:53.369 --> 00:16:00.178 Grows really fast is passed to the exponential and there's. 147 00:16:00.178 --> 00:16:05.938 Examples there is that sort of a gruesome example. 148 00:16:05.938 --> 00:16:14.009 I did gruesome examples from time to time. If you complain, I'll stop doing them, but let's imagine that we had 12 airplane crashes. 149 00:16:14.009 --> 00:16:19.979 Um, last year, which is fictitious who had 1 or something. 150 00:16:19.979 --> 00:16:33.509 What's the probability that the 12 crashes occurred in 12 different months? What's the probability that they were evenly spaced? 151 00:16:33.509 --> 00:16:36.599 And. 152 00:16:40.558 --> 00:16:44.519 Okay. 153 00:16:44.519 --> 00:16:47.729 Well, we've got basically. 154 00:16:49.979 --> 00:16:53.609 Crash. 155 00:16:53.609 --> 00:16:56.729 Which ones. 156 00:16:58.198 --> 00:17:03.389 Okay, so if they're all different and so how many raise. 157 00:17:05.249 --> 00:17:08.429 Silence. 158 00:17:08.429 --> 00:17:13.078 Well, this total factorial. 159 00:17:13.078 --> 00:17:18.419 Factorial ways if the crashes. 160 00:17:18.419 --> 00:17:23.278 In different months. 161 00:17:23.278 --> 00:17:26.699 In different months. 162 00:17:28.618 --> 00:17:31.919 Silence. 163 00:17:31.919 --> 00:17:35.969 As 12 to the 12, if we don't care. 164 00:17:38.368 --> 00:17:44.189 So, the probability of. 165 00:17:44.189 --> 00:17:50.608 The 12 crashes all in different months. 166 00:17:52.528 --> 00:17:56.068 That's going to be. 167 00:17:56.068 --> 00:18:04.648 121212, and that's going to be very small, have it on there. 168 00:18:05.788 --> 00:18:11.999 Okay, so it shows that random does not mean even. 169 00:18:11.999 --> 00:18:16.138 So, let's see if I can pull up. 170 00:18:16.138 --> 00:18:21.808 Good. That's better. Okay. I'll keep a little corner of the. 171 00:18:21.808 --> 00:18:26.459 Chat window open, so random does not mean evenly spaced. 172 00:18:28.378 --> 00:18:37.409 Okay, this happened without replacement or Nortel ordering is if you play poker and you're drawing a deck of cards hands. 173 00:18:37.409 --> 00:18:49.769 So, you're given cards from the deck the cards are not put back and you don't care what order you got the cards in. So, this is a natural example sampling. 174 00:18:49.769 --> 00:19:00.298 Without replacement, and without ordering, you just care what cards you'd get. You don't care what, what order you got them in. So that's combinations there. 175 00:19:00.298 --> 00:19:06.749 And that sad thing there reputation is and I did some of those examples here. 176 00:19:06.749 --> 00:19:11.368 Okay, and I did that. 177 00:19:13.499 --> 00:19:16.888 Multi nomial coefficients it's a generalization. 178 00:19:16.888 --> 00:19:20.548 Of the binomial coefficient, I'll call. 179 00:19:21.749 --> 00:19:26.368 I'll get that a little I want to move on to the book if you want to. Okay. 180 00:19:27.598 --> 00:19:32.759 Conditional probability. This is a big topic. 181 00:19:32.759 --> 00:19:36.568 And mathematically. 182 00:19:36.568 --> 00:19:43.078 I don't know, I actually don't play cards. I can't even deal with Decker cards, but it has. 183 00:19:43.078 --> 00:19:51.778 And I'm not expressing moral opinions on gambling and poker and dice. They just provide so many beautiful examples for this. Course. 184 00:19:51.778 --> 00:19:58.229 And also some of the mathematicians that are originally worked out probability. 185 00:19:58.229 --> 00:20:01.709 Had applications in mind involved in gambling. 186 00:20:01.709 --> 00:20:07.288 Okay, so a big thing is conditional probability is so. 187 00:20:07.288 --> 00:20:10.558 You know, you draw you look at your. 188 00:20:10.558 --> 00:20:18.568 You look at the and you've got and you want to know what's the probability that your opponents have certain hands? So. 189 00:20:18.568 --> 00:20:24.239 You could calculate the probability that somebody has a certain card ace of spades. If you do nothing at all. 190 00:20:24.239 --> 00:20:32.638 But you want to make conditioned on what you see what you know so that's conditional probability. 191 00:20:32.638 --> 00:20:37.318 And so if it's nice weather today, I'm sitting in my my. 192 00:20:37.318 --> 00:20:50.068 Office, I'm at home and routing, only 10 miles from RPI. I'm looking at the window. It's a bright, sunny day so conditioned on it being a bright shiny day. What's the probability? It might snow tomorrow. 193 00:20:50.068 --> 00:21:00.058 Or too low tonight is, I don't know what it's going to be. Let's say 15 Fahrenheit. What's the probability? The low tomorrow will be higher than that says conditional. 194 00:21:01.108 --> 00:21:09.808 Game stock went up today. I don't know if it did or not. What's the probability? Will go up to mark conditional probability we're going to hit that a lot. 195 00:21:09.808 --> 00:21:13.618 An independence of events are. 196 00:21:13.618 --> 00:21:17.219 If they don't affect each other. 197 00:21:17.219 --> 00:21:30.898 So, if knowing the temperature today does not affect temperature temperature tomorrow, then those would be independent events. But of course, it's not true. Temperature. Tomorrow will be similar to the temperature. Today. We'll see examples of that. 198 00:21:31.403 --> 00:21:44.334 And we get sequential experiments. We do something many times selling some technical terms. Binomial probabilities is we're tossing a sequence of coins. 199 00:21:44.634 --> 00:21:56.124 Let's say we have a coin and it's supposed to 1 and it comes up at 2 thirds of the time it's head. So, if we toss it 5 times, what's the probability? It'll get 3 heads let's say. 200 00:21:56.398 --> 00:22:04.528 So, binomial probabilities, multi nomial is that something came up here just a sec. 201 00:22:06.118 --> 00:22:10.769 There was a homework question. multinode. Okay. I'll do that. 202 00:22:10.769 --> 00:22:17.398 Okay, sure I'll do it with with the sequence of unfair dice. Let's say. 203 00:22:17.398 --> 00:22:20.878 If we. 204 00:22:20.878 --> 00:22:24.598 So, but it's got 6 paces maybe. 205 00:22:24.598 --> 00:22:30.868 What's the probability and maybe they've got different potbellies to the face is what's the probability. 206 00:22:30.868 --> 00:22:37.348 We see a certain thing. Okay, let me do a sample. 207 00:22:40.199 --> 00:22:44.578 Start a new pages here. 208 00:22:44.578 --> 00:22:51.568 4. 209 00:22:55.078 --> 00:22:59.969 Okay, I'm. 210 00:23:02.818 --> 00:23:09.898 I'll start with so I'll do binomial. 1st. Okay. 211 00:23:09.898 --> 00:23:13.199 Goodbye now me up 1st. 212 00:23:13.199 --> 00:23:16.888 And it was. 213 00:23:16.888 --> 00:23:19.949 Coins. 214 00:23:19.949 --> 00:23:27.148 I got 4 corn too. 215 00:23:28.528 --> 00:23:33.118 This 1 looks cool. 216 00:23:33.118 --> 00:23:37.739 This 1 is not an American penny. It's a. 217 00:23:40.108 --> 00:23:45.808 I don't know what it is, but it's not American any case. 4 times. 218 00:23:45.808 --> 00:23:50.189 And I don't know. 219 00:23:55.469 --> 00:23:58.679 Silence. 220 00:23:58.679 --> 00:24:02.308 Silence. 221 00:24:02.308 --> 00:24:07.108 Say, 2 sets. 222 00:24:08.398 --> 00:24:11.909 Of 2 well, I mean, basically. 223 00:24:13.108 --> 00:24:17.669 Raise to pick the 1st. 224 00:24:17.669 --> 00:24:24.088 The 2nd, 6, but divided by. 225 00:24:25.318 --> 00:24:29.548 Because 2, because each set. 226 00:24:29.548 --> 00:24:34.048 Pick 2 ways. 227 00:24:34.048 --> 00:24:42.209 Okay, so what we would have is and times and minus 1 and minus K plus 1. 228 00:24:42.209 --> 00:24:52.919 Divided by K factorial. The top 1 there is an factorial over in minus K choose. K. okay. 229 00:24:52.919 --> 00:25:02.368 Let me do 1 before I got a multi, let's do divide into. 230 00:25:06.118 --> 00:25:10.858 So That'll say and. 231 00:25:10.858 --> 00:25:18.179 Of 1, 1 element and so this is going to be and 2 to 3. 232 00:25:18.179 --> 00:25:31.439 which is a factor or effect zero one practical so for ways to divide into a set of three and a set of one set of one is any one of the four point the set of three . 233 00:25:31.439 --> 00:25:35.669 Is the other okay. 234 00:25:39.838 --> 00:25:47.548 Um. 235 00:25:47.548 --> 00:25:53.398 Okay, now I'm going to do not say I went back to the multi. 236 00:25:53.398 --> 00:25:58.739 Want to divide the coins. 237 00:25:58.739 --> 00:26:04.169 And to and just 3 sets. 238 00:26:04.169 --> 00:26:18.144 The size of 2, 1 and 1, and that would be the multi node. It'll be written as something like 4 choose 2 and 1 and 1. and That'll be factored over 2 factor. 1. 239 00:26:18.144 --> 00:26:20.784 factorial 1 factor equals 2. 240 00:26:22.108 --> 00:26:28.769 There's 2 ways 1 way to look at and these 3 sets are distinct. 241 00:26:28.769 --> 00:26:34.409 So, the 2 sets of size, mine are not interchangeable. Well, I could pick. 242 00:26:34.409 --> 00:26:41.909 Run under my gun here. 243 00:26:45.028 --> 00:26:50.429 Somebody who was awake could have gotten a point. 244 00:26:50.429 --> 00:26:54.239 12. 245 00:26:54.239 --> 00:26:57.838 Because let's suppose. 246 00:26:57.838 --> 00:27:09.209 The fair coins are quite a dime Nicole and Patty I could do a quarter and dime. 247 00:27:09.209 --> 00:27:12.959 I could do. 248 00:27:14.219 --> 00:27:22.648 Penny and Nicole, because I'm saying there's 2 such an audio changeable or creditor and penny. 249 00:27:22.648 --> 00:27:29.519 Nickel and dime are dime and Nicole or. 250 00:27:29.519 --> 00:27:37.798 Any and time quarter, any time corridor and Nicole, Nicole and corridor or. 251 00:27:38.848 --> 00:27:48.449 Nickel and dime penny in quarter quarter and a quarter dime quarter in any. 252 00:27:49.648 --> 00:28:01.318 For any and time time and Penny quarter and time quarter, quarter, a nickel nickel and dime. 253 00:28:05.669 --> 00:28:09.568 Nicole and Penny order and dime. 254 00:28:09.568 --> 00:28:13.019 Dial a quarter and that's going to be 12. 255 00:28:13.019 --> 00:28:18.808 So, it's. 256 00:28:18.808 --> 00:28:31.108 So this would be, I think the number of combinations of splitting something into 3 different, a different sets. Let me do a bigger example. 257 00:28:34.709 --> 00:28:39.088 Okay, let me see now. 258 00:28:40.888 --> 00:28:47.009 9, that's suppose we've got 9, 9 balls. 259 00:28:47.009 --> 00:28:59.368 And we want to and to 3 sets. 260 00:28:59.368 --> 00:29:05.398 Size 3 and 2. 261 00:29:06.419 --> 00:29:10.259 Number of ways. 262 00:29:10.259 --> 00:29:16.348 Come on you auto focus there. 263 00:29:16.348 --> 00:29:21.298 Okay, well. 264 00:29:21.298 --> 00:29:25.828 Set that will pick it up piece by piece the set up for. 265 00:29:25.828 --> 00:29:39.209 9 ways to pick the 1st ball 8 ways to pick the 2nd, 7 ways to pick the 3rd, 6 ways to pick the 4th divided by 24 ways because I could pick the same for about 24 different ways. 266 00:29:39.209 --> 00:29:47.159 So that's for the 1st. Sec, that would be 9 factorial over 4 factorial 5 bacterial. 267 00:29:48.898 --> 00:29:52.949 Ways to pick the 1st set. 268 00:29:52.949 --> 00:29:56.519 There's now, so there are. 269 00:29:58.138 --> 00:30:12.179 Files left to pick the 2nd set 5 ways to get this 1st Belfor for the 2nd, 3 for the 3rd divided by each 6 different ways to pick it. 270 00:30:12.179 --> 00:30:19.229 And that's factorial over 3 factor or 2 factorial. 271 00:30:19.229 --> 00:30:24.778 To pick the 2nd, 1 way to pick the 3rd. 272 00:30:24.778 --> 00:30:27.838 Silence. 273 00:30:27.838 --> 00:30:30.959 And you put it all together that's. 274 00:30:30.959 --> 00:30:41.729 9 factory over 4. factorial 5 factorial next 1 is 5 factorial 3 facto 2. factorial. And that will be 9 factor. 275 00:30:41.729 --> 00:30:48.419 Hi facto that covers 3 factorial written as 9. 5, 4 3. 276 00:30:48.419 --> 00:30:51.659 Or in tactile over. 277 00:30:51.659 --> 00:30:56.729 1 factor. Okay. 2 factorial factorial and so on. 278 00:30:56.729 --> 00:31:02.548 And you can prove it as it go through step by step you have. 279 00:31:02.548 --> 00:31:05.608 Yes, how did I get the following 1 the. 280 00:31:05.608 --> 00:31:08.999 4, 3, 2, instead of 5 3. 281 00:31:08.999 --> 00:31:12.868 Like, 5, 4, 3, the Victoria. 282 00:31:12.868 --> 00:31:16.288 We're on the page. 283 00:31:16.288 --> 00:31:21.179 Yeah, yeah, the, the, the right hand side of the equation. 284 00:31:21.179 --> 00:31:26.278 To be like 43254 3 I think. 285 00:31:26.278 --> 00:31:31.679 Material for and 3 and. 286 00:31:31.679 --> 00:31:35.578 So, is it because of. 287 00:31:35.578 --> 00:31:39.148 Yeah, you could email me you get a point. 288 00:31:39.148 --> 00:31:43.229 That was an on trivial error. Okay. 289 00:31:43.229 --> 00:31:49.679 You are correct so the 1st person email me and that would be probably, you. 290 00:31:49.679 --> 00:31:59.128 1st, whichever 1 emails me 1st gets the point. Okay. Thank you. Keep the errors come and keep the corrections coming. 291 00:31:59.128 --> 00:32:03.538 So that's okay. 292 00:32:04.979 --> 00:32:13.618 And where you don't care have the order in the SAT, you just care who's in the set. 293 00:32:17.398 --> 00:32:21.689 Okay. 294 00:32:21.689 --> 00:32:25.769 More questions would you like another example on that? 295 00:32:27.118 --> 00:32:30.358 It doesn't it has to add up to 9. yes, it does. 296 00:32:30.358 --> 00:32:33.898 Correct. Okay. 297 00:32:33.898 --> 00:32:38.999 So those and then, so that's the number of cases. 298 00:32:38.999 --> 00:32:42.898 And we can get multi nomial probabilities. 299 00:32:42.898 --> 00:32:48.538 Yeah, I'll work on. 300 00:32:48.538 --> 00:32:53.219 Do that. 301 00:32:53.219 --> 00:32:56.788 Page 7. 302 00:32:56.788 --> 00:33:00.719 Silence. 303 00:33:00.719 --> 00:33:04.108 No equipment has failed. Yeah. 304 00:33:04.108 --> 00:33:08.548 And going for half an hour, nothing has sailed. 305 00:33:09.989 --> 00:33:13.229 Know, what's going on? I'm not complaining. 306 00:33:13.229 --> 00:33:18.388 Okay. 307 00:33:18.388 --> 00:33:25.229 Oh, let's do a private bill a key now. 308 00:33:28.318 --> 00:33:34.979 I did some binomial probabilities. 309 00:33:36.209 --> 00:33:43.409 We got to get a fair call and then I'll make it a funny calling. 310 00:33:44.999 --> 00:33:49.528 And this notation probability of head this 1. 311 00:33:49.528 --> 00:33:54.689 Okay, so now we're going to start you just use the combination of things. 312 00:33:54.689 --> 00:34:01.528 Okay, and equals to. 313 00:34:01.528 --> 00:34:06.388 And then we get the probabilities. Okay. Probability. It had had. 314 00:34:06.388 --> 00:34:12.688 Equals 1 quarter probability of at kale had. 315 00:34:12.688 --> 00:34:15.688 Was 1 half and. 316 00:34:15.688 --> 00:34:20.009 Okay. 317 00:34:20.009 --> 00:34:27.028 And so number of ways to get. 318 00:34:27.028 --> 00:34:31.559 Exactly. 319 00:34:32.998 --> 00:34:37.349 Causes and choose K. 320 00:34:39.179 --> 00:34:44.938 Of course the number of outcomes. 321 00:34:46.829 --> 00:34:49.889 It was 2 to the end. 322 00:34:49.889 --> 00:34:53.998 So, the probability of exactly. 323 00:34:55.498 --> 00:35:01.168 8. 324 00:35:01.168 --> 00:35:04.648 And K, our 2 to the end. 325 00:35:06.688 --> 00:35:12.478 And that says an example here, and we tossed the calling 4 times. 326 00:35:12.478 --> 00:35:15.809 When I. 327 00:35:15.809 --> 00:35:19.739 And choose K is going to be. 328 00:35:19.739 --> 00:35:23.518 1, for 6 for 1. 329 00:35:23.518 --> 00:35:29.668 For the K cross 01234. 330 00:35:31.588 --> 00:35:35.878 So 6 ways. 331 00:35:35.878 --> 00:35:40.949 To get 2 heads no problem. 332 00:35:40.949 --> 00:35:45.778 Of 2 heads is going to be 6 over 16. 333 00:35:45.778 --> 00:35:49.318 Which 8. 334 00:35:50.458 --> 00:35:56.608 So, even though it's a fair calling, the probability of exactly. 2 heads is. 335 00:35:56.608 --> 00:36:05.608 3 8 is 6 ways we can get 2 heads out of the 16 things. Probably. Exactly. 2 heads. Okay. 336 00:36:05.608 --> 00:36:12.389 And so and then they'll say the 0T heads. 337 00:36:12.389 --> 00:36:19.349 Of a cake, or let's say that's going to be 116th and so on. That's fair point. 338 00:36:19.349 --> 00:36:22.438 Hello. 339 00:36:22.438 --> 00:36:36.148 I just my previous handwritten notes are up on. Okay. 340 00:36:36.148 --> 00:36:40.228 Kind. 341 00:36:40.228 --> 00:36:45.028 That's suppose we got the probability of. 342 00:36:45.028 --> 00:36:49.829 2 thirds of ahead. Okay. 343 00:36:49.829 --> 00:36:53.309 Okay, now. 344 00:36:53.309 --> 00:37:03.179 So, again, we're, we're going to task and times. 345 00:37:03.179 --> 00:37:07.409 And we see, hey, heads. 346 00:37:07.409 --> 00:37:11.878 Okay. 347 00:37:11.878 --> 00:37:15.059 So, the probability of. 348 00:37:15.059 --> 00:37:23.398 Let's say, oh, this task twice, we'll task twice and the probability we see no ads. 349 00:37:23.398 --> 00:37:29.099 Is. 350 00:37:29.099 --> 00:37:36.028 It's 4th times 4th, probably, the tail is 4th and that could be expressed. 351 00:37:36.028 --> 00:37:41.639 As the probably the care, which is 1 minus P. 352 00:37:41.639 --> 00:37:46.048 To the. 353 00:37:47.309 --> 00:37:50.369 To the end. Sorry. Okay. 354 00:37:50.369 --> 00:37:57.539 The probably of no. 355 00:37:58.588 --> 00:38:05.849 The probability of K, heads into some general and and. 356 00:38:05.849 --> 00:38:09.418 Well. 357 00:38:09.418 --> 00:38:14.070 Silence. 358 00:38:14.070 --> 00:38:19.170 Well, let me just look at a step by step. 359 00:38:19.170 --> 00:38:22.500 Okay, step 1 is the 1st part. 360 00:38:22.500 --> 00:38:31.260 So the probability of probably a K headset event. 361 00:38:31.260 --> 00:38:38.130 Hey, heads in a row. 362 00:38:38.130 --> 00:38:46.139 And minus K tails in a row. 363 00:38:46.139 --> 00:38:50.760 So the K headsets P2 the K. 364 00:38:50.760 --> 00:39:04.440 And then for the N minus K trail, that's probably the tail 1 minus P to the end minus. K. so this is the probability of K heads in a row and then N minus K tails in a row. 365 00:39:05.789 --> 00:39:09.989 Now, so the probability of K heads. 366 00:39:12.300 --> 00:39:22.920 Any order well, this is a probability of 1 or if I don't care about the news and choose K ways, this could happen. 367 00:39:22.920 --> 00:39:28.050 Times Peter the K times 1 minus speed at the end minus K. 368 00:39:28.050 --> 00:39:34.440 So this is a form where you're going to see a lot of the time. So, maybe I said. 369 00:39:34.440 --> 00:39:38.400 You know, maybe, I don't know. 370 00:39:38.400 --> 00:39:44.670 Yeah, sort of thing you're going to see a lot of the time. 371 00:39:44.670 --> 00:39:48.150 The probability of key heads out of and. 372 00:39:48.150 --> 00:39:55.710 I don't care about the order and it's unfair to search and P, chance of getting ahead. 373 00:39:55.710 --> 00:40:01.980 You see, so what's P and 1 minus P the piece the probability of getting ahead. 374 00:40:03.000 --> 00:40:09.989 Which, in this case, it'd be 2 thirds. So, because 1 minus P is the probability of getting a tail. 375 00:40:09.989 --> 00:40:15.989 Sam. 376 00:40:15.989 --> 00:40:23.489 Even tail and 1 toss 1 minus. P. okay. 377 00:40:24.750 --> 00:40:30.059 Okay, so. 378 00:40:30.059 --> 00:40:38.940 So, this binomial okay and now I'm getting sad is pointy here. Binomial probabilities. 379 00:40:40.199 --> 00:40:43.320 And specifically. 380 00:40:43.320 --> 00:40:48.269 Do an example, work out some things that's suppose. 381 00:40:48.269 --> 00:40:52.860 So some numbers. 382 00:40:52.860 --> 00:41:04.650 And equals 4 K cause to another peak was 2 thirds. So the probability of exactly. 2 heads out of the. 383 00:41:04.650 --> 00:41:10.980 Process each head is 2 thirds and this will be 4 Jews 2. 384 00:41:10.980 --> 00:41:16.800 Times Square times 4th Square. 385 00:41:17.849 --> 00:41:26.579 And that's going to be 6 times for over 8124 over 81. 386 00:41:27.840 --> 00:41:31.860 And do whatever, you know. 387 00:41:31.860 --> 00:41:34.860 Give or take a quarter or something so. 388 00:41:38.429 --> 00:41:43.679 It's about point 3.3 roughly. 389 00:41:43.679 --> 00:41:50.400 Okay. 390 00:41:50.400 --> 00:41:57.929 Any calls in K cause I believe all 4 heads, then that's going to be 2 thirds to the 4th. 391 00:41:57.929 --> 00:42:00.960 And that's going to be. 392 00:42:00.960 --> 00:42:06.059 About point 2 or something. Oh, okay. 393 00:42:07.650 --> 00:42:11.099 Probabilities. 394 00:42:12.239 --> 00:42:18.300 The question, so this is when it's an unfair coin. 395 00:42:18.300 --> 00:42:27.150 Yes, it is an unfair coin. Yes, we need to talk about claims. We could talk about your building widgets on the assembly line. 396 00:42:27.150 --> 00:42:31.469 And the probability that it's a bad widget is maybe 1%. 397 00:42:31.469 --> 00:42:37.739 Okay, so piece point 1 and now that the probability that out of 10 widgets, 2 of them are bad. 398 00:42:37.739 --> 00:42:42.150 It's something like that. 399 00:42:52.019 --> 00:42:58.170 So, an example. 400 00:42:58.170 --> 00:43:05.190 Just. 401 00:43:05.190 --> 00:43:08.280 Equals 1. 402 00:43:08.280 --> 00:43:15.119 It is bad, so I wrote the probability that. 403 00:43:15.119 --> 00:43:20.039 I don't know, you know, to of can reject. 404 00:43:20.039 --> 00:43:23.550 I bet okay. 405 00:43:23.550 --> 00:43:29.159 Well, that's going to be 10 choose 2. 406 00:43:29.159 --> 00:43:33.389 And the probability of bad point 1. 407 00:43:33.389 --> 00:43:37.289 Squared times point 9, 9 to the. 408 00:43:38.969 --> 00:43:44.909 Okay, the problem exactly to the 10 widgets are bad. If 1, which is a 1% chance. 409 00:43:44.909 --> 00:43:55.289 Okay, Eva. Okay. 410 00:43:56.489 --> 00:44:08.309 Multi nomial fair coin. I just saw that. Yeah, but you may want the. 411 00:44:08.309 --> 00:44:14.909 Chance yeah, you may want to know what's the chance you'll get exactly. 412 00:44:16.920 --> 00:44:28.530 Let's see either for clients or heads or something, you know, you're having kids, your parents are having kids and you got 2 kids. What's the probability to each 1? Boy 1 girl. 413 00:44:28.530 --> 00:44:38.219 And so on, or you've got 4 kids, what's the probability? There'll be 3 boys and 1 girl or something so where we're sending the probabilities 50, 50. 414 00:44:38.219 --> 00:44:41.369 Okay, do you know me. 415 00:44:42.539 --> 00:44:45.780 You know. 416 00:44:47.820 --> 00:44:52.920 What would be an example. 417 00:44:59.190 --> 00:45:05.699 I got this. 418 00:45:05.699 --> 00:45:10.710 Thing I got it at a conference for free. It's from. 419 00:45:12.239 --> 00:45:17.309 Oh, okay. 420 00:45:17.309 --> 00:45:24.690 So, I'm going to have probability of selecting a color. 421 00:45:24.690 --> 00:45:28.050 Okay, um. 422 00:45:32.010 --> 00:45:41.219 And since me. 423 00:45:42.929 --> 00:45:46.050 Hello. 424 00:45:46.050 --> 00:45:49.679 Okay, and that's mark on. 425 00:45:49.679 --> 00:45:57.630 This is not working very well at the bank. 426 00:45:57.630 --> 00:46:02.550 Hello. 427 00:46:02.550 --> 00:46:06.869 That worked better. 428 00:46:06.869 --> 00:46:09.929 Nicely. 429 00:46:12.210 --> 00:46:17.880 Sort of. 430 00:46:19.769 --> 00:46:24.630 And orange well. 431 00:46:24.630 --> 00:46:29.099 Label it that is actually orange. 432 00:46:29.099 --> 00:46:32.280 Oh, okay. 433 00:46:32.280 --> 00:46:35.340 Orange, thank. 434 00:46:35.340 --> 00:46:40.650 Yellow green blue. 435 00:46:40.650 --> 00:46:43.889 Okay, so basically. 436 00:46:43.889 --> 00:46:47.460 I'm picking a color at random. 437 00:46:50.010 --> 00:47:01.559 And I'm like, and let's suppose the probability of picking orange is point Ron. 438 00:47:01.559 --> 00:47:06.539 Bank is 2.3. 439 00:47:06.539 --> 00:47:10.349 What am I at that point? 6. 440 00:47:13.769 --> 00:47:26.550 Green is 3 and blue is point 1. I'll make them all different number 1.1 or something. Okay so. 441 00:47:28.619 --> 00:47:32.969 Stand up to 1 if I did it, right? Yeah. Okay. So. 442 00:47:35.909 --> 00:47:43.469 3, Carlos. 443 00:47:45.510 --> 00:47:54.179 You know, without replacement or something, so that would be. 444 00:48:06.269 --> 00:48:11.190 So that's suppose I. 445 00:48:11.190 --> 00:48:14.280 Silence. 446 00:48:14.280 --> 00:48:21.719 Okay, how would we do this? Not. 447 00:48:21.719 --> 00:48:27.659 Heck, 10 colors. Okay. 448 00:48:27.659 --> 00:48:32.250 The probability is that they're all blue. 449 00:48:32.250 --> 00:48:39.119 Equals point 1 because it okay. 450 00:48:39.119 --> 00:48:43.349 I am. 451 00:48:43.349 --> 00:48:48.360 The probability of the 10 Carlos. 452 00:48:48.360 --> 00:48:51.900 That there are 2 blue. 453 00:48:51.900 --> 00:48:55.289 2 green. 454 00:48:58.590 --> 00:49:01.800 And. 455 00:49:04.590 --> 00:49:11.400 The same to bank and 2, orange. 456 00:49:11.400 --> 00:49:20.369 Well, probability it's going to be 10 choose to. 457 00:49:20.369 --> 00:49:24.420 Times are. 458 00:49:24.420 --> 00:49:35.849 The new point, 1 squared the green point. 3 Square, the yellow point 3 squared the pink point. 2 squared and the orange point. 1 squared. 459 00:49:35.849 --> 00:49:39.329 Because this is the way here. 460 00:49:39.329 --> 00:49:49.289 On the right that I would pick to blue green to add that to pink then 2 orange and this is a number of different ways. 461 00:49:49.289 --> 00:49:56.340 I can do that and that, by the way we over to the fact. 462 00:49:56.340 --> 00:50:01.889 Okay, something like that. So that's a multi nomial probability. 463 00:50:01.889 --> 00:50:06.329 Say. 464 00:50:06.329 --> 00:50:11.340 And. 465 00:50:12.869 --> 00:50:16.739 So, do you care what the set is, but you don't care. 466 00:50:16.739 --> 00:50:19.860 Hello. 467 00:50:19.860 --> 00:50:24.989 Do you okay? The already you draw it? You're playing a card game. 468 00:50:24.989 --> 00:50:29.130 And you care about how many hearts and. 469 00:50:31.500 --> 00:50:34.739 Well, this is so. 470 00:50:34.739 --> 00:50:41.940 The number of hearts, the number of clubs and so on, you've got, but you don't care the order that you drew them in. So. 471 00:50:41.940 --> 00:50:47.940 Let me do that. 472 00:50:47.940 --> 00:50:51.539 Okay. 473 00:50:56.550 --> 00:50:59.639 Silence. 474 00:51:00.840 --> 00:51:06.150 Saying a funny card game. 475 00:51:06.150 --> 00:51:10.320 And there's. 476 00:51:11.699 --> 00:51:16.320 Silence. 477 00:51:18.570 --> 00:51:23.969 However, there are, um. 478 00:51:26.550 --> 00:51:30.900 I don't know. 479 00:51:30.900 --> 00:51:34.920 There's 15 hearts. 480 00:51:34.920 --> 00:51:38.489 20 clubs, um. 481 00:51:38.489 --> 00:51:43.889 10 spades club spades. 482 00:51:43.889 --> 00:51:51.119 Can move that the window does cover part of it. 483 00:51:51.119 --> 00:51:54.570 How's that? Thank you. 484 00:51:56.639 --> 00:52:00.960 What's the 4th, so I don't play cards. 485 00:52:06.449 --> 00:52:10.980 Anyone diamonds, 30, 45. 486 00:52:10.980 --> 00:52:14.429 7 diamonds, it's a really okay. 487 00:52:15.750 --> 00:52:23.730 Okay, so now, so the problem 1, heart, of course, is going to be. 488 00:52:23.730 --> 00:52:33.900 1552, so now, as far as I want the probability that I draw, I hand of. 489 00:52:33.900 --> 00:52:39.389 2 hearts and 1 club. 490 00:52:39.389 --> 00:52:45.150 And, I don't know, 2 diamonds. Okay. 491 00:52:47.699 --> 00:52:52.230 Okay, so what's this? So, 1st, a number of ways I can do this. 492 00:52:52.230 --> 00:52:55.949 Will be 5 choose 2, 1 and 2. 493 00:52:55.949 --> 00:53:03.989 And too hard so probably the heart is 1550 seconds. So if gene 50 seconds squared 1 club. 494 00:53:03.989 --> 00:53:09.360 It's 20, 50 seconds and 1. 495 00:53:09.360 --> 00:53:12.719 A diamond is 750 seconds. 496 00:53:12.719 --> 00:53:16.860 Squared so that's the probability in this. 497 00:53:16.860 --> 00:53:22.619 Crazy card deck. I'm drawing a 5 card hand of 2 hearts a club and 2 diamonds. 498 00:53:22.619 --> 00:53:28.500 Well, this is an on ref of each suit in that 52 card deck. 499 00:53:28.500 --> 00:53:33.150 Okay, can you show the. 500 00:53:33.150 --> 00:53:38.429 2 of the 5 choose to attention 1 inches to that. 501 00:53:38.429 --> 00:53:44.489 I'm sorry could you say that again? You can you show like. 502 00:53:44.489 --> 00:53:50.039 What was the exact calculation inside the. 503 00:53:50.039 --> 00:53:55.920 12 5 truth to enter the 1 to 2. sure. Okay. Operate. Yes. 504 00:54:01.650 --> 00:54:06.570 This means. 505 00:54:06.570 --> 00:54:15.780 Okay, so the way you pronounce that is maybe 5 choose 2 and then 1 and then 2. 506 00:54:15.780 --> 00:54:22.500 And it's and the formula is 5 page. 507 00:54:22.500 --> 00:54:26.340 There we go get the chat window out. 508 00:54:27.900 --> 00:54:33.360 On Thank you diamonds. Yeah. 509 00:54:35.460 --> 00:54:39.389 Can put it on top like that. Okay. Okay. 510 00:54:40.949 --> 00:54:46.559 Okay on. 511 00:54:47.610 --> 00:54:50.610 What it means is that. 512 00:54:52.110 --> 00:54:56.550 It's a number of ways. 513 00:54:56.550 --> 00:55:01.619 Hi. 514 00:55:01.619 --> 00:55:05.280 I set up 5 things. 515 00:55:07.469 --> 00:55:13.260 News are into. 516 00:55:13.260 --> 00:55:19.650 A set of 2 and a set of 1. 517 00:55:19.650 --> 00:55:25.019 And a set of 1, and the way you prove it. 518 00:55:25.019 --> 00:55:33.449 Is find a set of 2. 519 00:55:33.449 --> 00:55:38.340 5 ways to. 520 00:55:38.340 --> 00:55:46.889 The 1st thing times for the ways to pick the 2nd thing. 521 00:55:48.090 --> 00:55:55.920 I would have to divide by 2 because each set could we picked 2 ways. 522 00:55:55.920 --> 00:55:59.489 Silence. 523 00:56:02.429 --> 00:56:05.969 Have a good question. Sure. 524 00:56:05.969 --> 00:56:10.710 For the end 1 and 1 are those interchangeable. 525 00:56:12.539 --> 00:56:15.809 No, they are not. 526 00:56:15.809 --> 00:56:22.110 So, 2, 1, 2, and it should be a 1 here. 2, 1 and 2. yeah. I mean, it was 5 so. 527 00:56:22.110 --> 00:56:29.429 That's a good point. If they were to changeable. That would be a perfectly legitimate problem, but it's a different problem. 528 00:56:29.429 --> 00:56:33.869 So, and it depends what your application is. 529 00:56:33.869 --> 00:56:41.039 So, Eva. 530 00:56:41.039 --> 00:56:48.059 Example of this distributing cards around the dealer and you distributing cards around the. 531 00:56:48.059 --> 00:56:53.670 Around the table, so the number of ways you can get this person maybe. 532 00:56:53.670 --> 00:56:57.659 To diamonds and experts in 1 and the 3rd person or something. 533 00:57:00.030 --> 00:57:04.230 Okay, Mark more multi nomial questions. 534 00:57:06.480 --> 00:57:10.230 I have a question on that. Why why do you mind like, 2. 535 00:57:10.230 --> 00:57:15.269 Let me give an example to show it to. 536 00:57:15.269 --> 00:57:19.650 Let me see. 537 00:57:19.650 --> 00:57:27.449 Focus feature on the cameras. Weird. Oh, okay. 538 00:57:27.449 --> 00:57:32.099 So 5 things. 539 00:57:32.099 --> 00:57:37.230 A B. C. 540 00:57:37.230 --> 00:57:44.190 Okay um, so there are. 541 00:57:45.630 --> 00:57:49.440 Ways to pick. 542 00:57:49.440 --> 00:57:56.099 A set of 210 sets. 543 00:57:56.099 --> 00:57:59.159 Up to. 544 00:57:59.159 --> 00:58:07.679 From that, because I could have a with B. C. D. E. 545 00:58:07.679 --> 00:58:11.280 I could be with C. D. 546 00:58:11.280 --> 00:58:24.269 See with D. R. E. this means any 1 of those 5 B. they could have any 1 of those 3 C. could have and take only be for 3 to 1 that equals 10. 547 00:58:26.610 --> 00:58:31.079 So there are 10 ways to pick. 548 00:58:31.079 --> 00:58:40.469 2 letters from that set of 5 where I count what's in that set up 2 letters, but I don't care the order. I picked them. 549 00:58:41.849 --> 00:58:46.230 So, if I'm picking coins. 550 00:58:46.230 --> 00:58:51.539 5 coins that's 4. 551 00:58:59.039 --> 00:59:03.599 Hi, okay. This is a. 552 00:59:06.420 --> 00:59:13.199 Is it is it's a dollar coin. Okay. So. 553 00:59:13.199 --> 00:59:21.150 5 coins here, put them in order value a dollar 25 cents, 10 cents 5 cents and 1 cent. 554 00:59:21.150 --> 00:59:29.219 Okay, so how many different ballances can I get if I pick this? 1 and this 1 I got a dollar 2005. 555 00:59:29.219 --> 00:59:44.070 If I pick this 1 and this 1, I got a dollar and 5. so, you see, I care the 2 coins in my set, but I don't care the order I get them in because all I care is how much money I've got at the end of it. 556 00:59:45.150 --> 00:59:52.530 So so, okay, so this is a combination choosing to. 557 00:59:52.530 --> 00:59:58.920 From the set of 5. okay. And now. 558 01:00:00.960 --> 01:00:03.960 Maybe, um. 559 01:00:06.750 --> 01:00:13.349 I don't know well, I could extend this example if you want, but this is a case. 560 01:00:13.349 --> 01:00:17.250 Where I care what the set I pick is. 561 01:00:17.250 --> 01:00:20.909 But I don't care the order. I pick the things in the set. 562 01:00:23.070 --> 01:00:26.280 Okay, mathematically. 563 01:00:26.280 --> 01:00:29.940 Why is there a division with 2? 564 01:00:29.940 --> 01:00:33.900 Because the way I would, I would get this. 565 01:00:33.900 --> 01:00:37.800 Is, um. 566 01:00:37.800 --> 01:00:43.679 5 ways to pick the 1st coined. 567 01:00:43.679 --> 01:00:52.590 Times 4 ways to pick the 2nd, because it's only 4 coins left. That's 20. but each set could get picked 2 ways. So. 568 01:00:52.590 --> 01:00:57.780 It's divided by 2 ways. 569 01:00:57.780 --> 01:01:06.000 And he said, so, for example, if I pick a dollar and a quarter. 570 01:01:06.000 --> 01:01:18.179 Okay, so you see 20 things here, I could pick a dollar in a quarter, or I could pick a quarter and then a dollar. Those are 2 different things. If I'm picking. 571 01:01:18.179 --> 01:01:26.670 You know, it'd be 20 different things, quarter dollars, etc or dollar quarter, but I don't care if you divide by 2. 572 01:01:28.889 --> 01:01:34.409 I just want to make a note that too is equal to 2 factorial in this case, which is like yeah, right. Yeah. 573 01:01:37.349 --> 01:01:45.059 Um, well, let me do it. I'll do a smaller example, because I want to nail down the basics. 574 01:01:48.360 --> 01:01:52.050 Page 13. 575 01:01:52.050 --> 01:01:56.340 Pick 2 coins. 576 01:01:56.340 --> 01:02:04.650 Come 3 such writing are going to send 2 letters. 577 01:02:06.090 --> 01:02:14.190 Okay, so a B. C. so here's all the ways I could pick this. I could pick a and then B a. and then see. 578 01:02:14.190 --> 01:02:17.610 B, because it's without replace, but I can't pick twice. 579 01:02:17.610 --> 01:02:21.750 In an, a. B and C. C. and then a and C and B. 580 01:02:21.750 --> 01:02:25.440 That's 6. okay. Okay. 581 01:02:25.440 --> 01:02:30.000 But if I care what the thing, here's the thing is that. 582 01:02:30.000 --> 01:02:33.360 You know, a B and B a. 583 01:02:33.360 --> 01:02:37.860 You know, that's the same set and AC. 584 01:02:37.860 --> 01:02:41.159 And C a, is the same set and so on. 585 01:02:42.179 --> 01:02:46.050 Okay, so. 586 01:02:46.050 --> 01:02:50.309 Said. 587 01:02:51.420 --> 01:02:54.420 I speak twice. 588 01:02:54.420 --> 01:02:58.110 So so that's why we. 589 01:02:58.110 --> 01:03:01.469 So, we. 590 01:03:01.469 --> 01:03:05.309 By 2 and 3. 591 01:03:05.309 --> 01:03:09.900 And so the 3 sets. 592 01:03:11.940 --> 01:03:20.909 You can pick 3 sets of 2 elements. You can pick from the 3 letters. 593 01:03:22.260 --> 01:03:27.329 Silence. 594 01:03:32.579 --> 01:03:39.179 So, if you care about the order, that's 6, if you don't care about the order that's 3. 595 01:03:39.179 --> 01:03:44.250 And the 6 is, um, is this an factorial. 596 01:03:44.250 --> 01:03:49.829 Divided by and minus K pack and equals 3. K equals 2. yeah. 597 01:03:49.829 --> 01:03:55.079 And we divide through by 2, we're dividing by K factorial. 598 01:03:55.079 --> 01:04:00.449 Which is defined as and. 599 01:04:00.449 --> 01:04:04.650 And choose K. 600 01:04:09.840 --> 01:04:17.789 Side the note here. 601 01:04:19.679 --> 01:04:26.909 Right, yeah, to the left. So you can see that. 602 01:04:26.909 --> 01:04:34.559 Yeah, more questions. 603 01:04:46.199 --> 01:04:53.550 Maybe I should do another example. 604 01:04:53.550 --> 01:04:57.929 Let's see here. 605 01:05:02.429 --> 01:05:08.699 Well, well, wait and see if you can think about that way to see it's another exact question. 606 01:05:17.039 --> 01:05:20.940 Do another example with. 607 01:05:20.940 --> 01:05:25.110 14. 608 01:05:25.110 --> 01:05:31.139 Let's do I'm going to die. 609 01:05:32.219 --> 01:05:37.170 Excited I. 610 01:05:37.170 --> 01:05:41.639 Okay case 1 it's fair. 611 01:05:41.639 --> 01:05:54.239 Okay, and so the probability say of coming up with a 1 equals a probably better comment with a 2 equals 1. 6. okay. So. 612 01:06:01.170 --> 01:06:06.809 I don't know, let's suppose that's 10 times. 613 01:06:08.789 --> 01:06:11.969 So the probability and let's say. 614 01:06:12.989 --> 01:06:16.050 Of 1, 1 to 2. 615 01:06:16.050 --> 01:06:21.750 33506 says. 616 01:06:21.750 --> 01:06:25.110 What this means is. 617 01:06:25.110 --> 01:06:28.469 The 3 came up 3 times, so. 618 01:06:28.469 --> 01:06:32.010 So, he came up 3 times. 619 01:06:32.010 --> 01:06:39.690 5 came up 0T times, so probably, it's that. So we've got the multi nomial of. 620 01:06:42.719 --> 01:06:45.750 Can choose. 621 01:06:47.429 --> 01:06:52.889 12340. 622 01:06:52.889 --> 01:06:59.730 Times a probability, which will be 1 6 to the 10th. 623 01:06:59.730 --> 01:07:03.329 So this is a probability of exactly. 624 01:07:03.329 --> 01:07:13.380 When the guy coming up the 1 coming up, once the 2 coming up, twice the 3 coming up 3 times before coming up 4 times in the 5 and 6 coming at not at all. 625 01:07:13.380 --> 01:07:27.000 For the fair die and this thing here is 10 fact from over 1 factor up to 4 factorial 0T factorial 0T factorial. 626 01:07:27.000 --> 01:07:31.679 So now. 627 01:07:33.329 --> 01:07:41.610 Let's say it's an unfair die. So case 2 now it's unfair. 628 01:07:41.610 --> 01:07:45.360 I don't know the. 629 01:07:48.329 --> 01:07:52.050 How could we make the thing unfair? 630 01:07:55.110 --> 01:08:00.210 The 1, 1. 631 01:08:00.210 --> 01:08:03.690 Is now, let's say, 50%. 632 01:08:03.690 --> 01:08:08.820 The point 5 to. 633 01:08:08.820 --> 01:08:11.969 I mean, because a bit of a 3, et cetera. 634 01:08:11.969 --> 01:08:23.399 To probably 6 equals point 1 and adds up to 1. good. So, now the probability of getting 1 run 2, 2, 3, threes and this and That'll be. 635 01:08:23.399 --> 01:08:28.079 Steadily number combinations 12340 0. 636 01:08:28.079 --> 01:08:40.470 Now, we've got we've got the run run, so That'll be a point. 5 the 2 is a point 1. so be point 1 squared point 1 2.124.120. 637 01:08:40.470 --> 01:08:46.380 Point 1 to the 0T. So this is the probability of this combo happening. 638 01:08:46.380 --> 01:08:52.649 When the 1 face has a 50% chance, and each of the other 5 cases is a 10% chance. 639 01:08:55.109 --> 01:09:03.329 Silence. 640 01:09:06.960 --> 01:09:13.949 Right, exactly. The basically question is correct. Yes, it makes the writing easier. 641 01:09:14.970 --> 01:09:18.329 Some bucks of different ways to write it, but this is the common 1. 642 01:09:22.619 --> 01:09:35.970 Okay, everyone's totally started. This is enough stuff for today Thursday I'll move on to other stopped geometric and so on get into conditional. That's your big topic. 643 01:09:35.970 --> 01:09:41.220 And so what I did today was just as. 644 01:09:41.220 --> 01:09:46.859 Stuff, how is it what is going on with homework? 3. 645 01:09:46.859 --> 01:09:52.199 We're going to do little homeworks based on what actually happened in the lecture and so on. 646 01:09:52.199 --> 01:09:58.319 So, There'll be small, only a few, and There'll be multiple choice questions. So we can grade them on. 647 01:09:58.319 --> 01:10:03.840 On grade scope, so. 648 01:10:08.520 --> 01:10:12.569 And, okay, good. 649 01:10:12.569 --> 01:10:16.710 Yeah, he is. Don't give you a chance to forget the stuff. 650 01:10:16.710 --> 01:10:20.609 No, other questions. 651 01:10:22.319 --> 01:10:27.689 Right because we like 5 answers. So if you get to pick the right answers so that'd be simple. 652 01:10:29.340 --> 01:10:37.170 When will it get posted? Real soon now, the teaching assistants I'm going to take turns creating them and sapping them on. Great. Scopes. 653 01:10:38.460 --> 01:10:47.159 It's already off. Oh, okay. Other questions and. 654 01:10:49.500 --> 01:10:54.000 Okay, I work for. Okay. 655 01:10:55.140 --> 01:10:59.939 1st, couple of homeworks are bigger tricking to to more frequent, little ones. So. 656 01:11:01.710 --> 01:11:08.970 Other thing is a, you know, the way I find our best what's happening in other classes. 657 01:11:08.970 --> 01:11:14.550 It's when you guys tell me, so it's a, it's a good technique that's in another class. 658 01:11:14.550 --> 01:11:20.850 Tell me, and maybe perhaps I might use it. No guarantees, but I'd like to know what's happening any other courses. 659 01:11:22.050 --> 01:11:28.710 Because, you know, we do want to, like I said, we want you guys to succeed. 660 01:11:28.710 --> 01:11:33.390 Because it makes us look good for no other reason. And. 661 01:11:33.390 --> 01:11:37.529 That doesn't mean to give everyone a automatically. 662 01:11:37.529 --> 01:11:40.770 You know, if you're going out and building a bridge, we don't want it to fall down. 663 01:11:40.770 --> 01:11:46.229 So, you know, there was an RPI grad. 664 01:11:46.229 --> 01:11:52.079 Many many years ago who built a big bridge and a collapse and a killed 80 people. 665 01:11:52.079 --> 01:11:57.149 And we named a dorm after him, his name was Cooper. 666 01:11:57.149 --> 01:12:01.020 So don't be like Cooper place. 667 01:12:01.020 --> 01:12:05.039 Ah, okay. 668 01:12:05.039 --> 01:12:08.460 And show you the bridge to Southern grad. 669 01:12:08.460 --> 01:12:12.090 Silence. 670 01:12:12.090 --> 01:12:19.020 Silence. 671 01:12:22.380 --> 01:12:28.079 Yeah, bridges aren't supposed to look like that. Okay. 672 01:12:28.079 --> 01:12:32.130 Not at all and, um. 673 01:12:34.199 --> 01:12:40.079 Who to do the consulting engineer. 674 01:12:44.039 --> 01:12:54.989 Don't be like Cooper on the other hand, I had to go back a really long time to find that example. 675 01:12:54.989 --> 01:12:59.340 Because most of you guys are really good. Okay that's why we want you to learn this stuff. 676 01:13:01.710 --> 01:13:06.239 It says to all more questions. See, you Thursday. 677 01:13:12.000 --> 01:13:15.029 I ask a question about the the homework too. 678 01:13:15.029 --> 01:13:18.239 Sure, let me pull it up. 679 01:13:22.140 --> 01:13:29.970 Silence. 680 01:13:32.340 --> 01:13:43.409 Question 6 sure. Let me pull it up. 681 01:13:47.460 --> 01:13:50.909 Silence. 682 01:13:50.909 --> 01:13:55.140 Silence. 683 01:14:01.829 --> 01:14:16.140 Silence. 684 01:14:18.329 --> 01:14:24.239 You know, okay, let me drop then. 685 01:14:24.239 --> 01:14:29.159 I'll set up the, I can put up the screen again. 686 01:14:32.130 --> 01:14:38.789 The 2nd. 687 01:14:38.789 --> 01:14:52.050 Silence this may not work. 688 01:14:53.789 --> 01:14:59.399 I'm not certain I can start the, the web cam again. 689 01:14:59.399 --> 01:15:07.890 So, Eva. 690 01:15:12.149 --> 01:15:15.449 Okay. 691 01:15:17.430 --> 01:15:21.720 Okay, so. 692 01:15:37.289 --> 01:15:43.380 Well, I guess it could be interpreted 2 different ways, but. 693 01:15:43.380 --> 01:15:50.699 Half is frequently now, so the probability of the number occurring, and some Delta X. 694 01:15:50.699 --> 01:15:56.010 Problems there, and there are 2 and a half as much as in the minus 1 to 0T and are fall. 695 01:15:56.010 --> 01:16:01.739 So, the. 696 01:16:06.270 --> 01:16:17.220 I would I'm looking at something like a quarter to be interval minus 1, because they're probably in any little if they say half is frequently. 697 01:16:19.890 --> 01:16:23.039 Yeah, I'm saying this could actually be. 698 01:16:23.039 --> 01:16:27.689 Imagine 2 different ways so, in a case like that, it's. 699 01:16:27.689 --> 01:16:31.229 Date and assumption, and then. 700 01:16:31.229 --> 01:16:34.619 Work from your assumption, what assumption. 701 01:16:34.619 --> 01:16:44.670 Is that the whole interval? 0 2 the probably if anything doesn't whole interval is half as frequently as anything in the whole interval minus 1. 0T in which case the answer to a. 702 01:16:44.670 --> 01:16:47.670 Is 4th to the minus 1? 0. 703 01:16:47.670 --> 01:16:54.569 You and 4th. 704 01:16:54.569 --> 01:17:00.119 Yeah. 705 01:17:00.119 --> 01:17:03.840 Well, 4th to the minus 1. 706 01:17:04.949 --> 01:17:07.979 1, so, for those are the 2, it's the whole goal is. 707 01:17:07.979 --> 01:17:14.250 Half is half as frequently, but another way it could be interpreted as any fixed little length. 708 01:17:14.250 --> 01:17:24.090 Is half as frequently as the same fixed little length in minus 1 0T so it could be argued. That would also. 709 01:17:24.090 --> 01:17:28.649 Is a reasonable reading of the question so. 710 01:17:33.569 --> 01:17:39.359 And part in each of the above intervals, it's a number it's not an interval. So. 711 01:17:43.560 --> 01:17:49.739 Heart needs of the above intervals is 0T because it's for an interval. 712 01:17:49.739 --> 01:17:53.220 I should have read this in more detail. 713 01:18:07.979 --> 01:18:16.590 Silence. 714 01:18:22.500 --> 01:18:30.119 Yeah, okay. You will email me about this. What I'm going to do is delete the question. 715 01:18:32.369 --> 01:18:35.850 So, because. 716 01:18:35.850 --> 01:18:43.140 It's bad I should have read it more slowly when I was picking it. So email me and. 717 01:18:43.140 --> 01:18:47.100 1st price to email me gets a point about this and. 718 01:18:47.100 --> 01:18:50.310 Well, I will delete the question and not greater. 719 01:18:50.310 --> 01:18:54.359 Yeah. 720 01:18:54.359 --> 01:18:58.109 I also have a question on. 721 01:18:58.109 --> 01:19:01.680 8, 6. 722 01:19:03.569 --> 01:19:17.159 Silence. 723 01:19:20.670 --> 01:19:35.545 Side is the multi nomial thing that we've been doing. 724 01:19:35.604 --> 01:19:35.965 Yeah. 725 01:19:40.050 --> 01:19:44.220 Of 21 students each through a bar in each day of the week. So. 726 01:19:44.220 --> 01:19:52.079 What's the question? So easy. 727 01:19:58.770 --> 01:20:03.449 20 and 21 students. Exactly. 3 reward in each of the 7 days of the week. 728 01:20:03.449 --> 01:20:22.079 Silence. 729 01:20:22.079 --> 01:20:26.100 It's a multi normally 21 2 303 7 threes. 730 01:20:26.100 --> 01:20:35.729 And they divide by 7 to the 21 to probability of any 1 day. 731 01:20:35.729 --> 01:20:39.779 Is 1, 7th and there's 21 so that's that. 732 01:20:39.779 --> 01:20:44.579 You know, Peter, the end thing that divide by 7 to the 21. 733 01:20:48.300 --> 01:20:51.510 You see the divide by 7 by. 734 01:20:51.510 --> 01:20:56.039 The divide by 7 to the 2001 is because that's the total. 735 01:20:56.039 --> 01:21:10.079 Well, that's that multi normally probability thing. We've got that combination term like, and to choose K1, K2 k4 times the probability 1 to the K1 time speed to the K2. 736 01:21:10.079 --> 01:21:13.949 I speak to the K3 and so on and. 737 01:21:13.949 --> 01:21:19.319 Well, each of the piece here is 1, 7th, and all of the case that just 21 students. 738 01:21:20.430 --> 01:21:26.880 That's dividing by 7 to the 21. okay. 739 01:21:32.789 --> 01:21:36.359 Does that make sense or not? And. 740 01:21:37.439 --> 01:21:42.359 I don't know it doesn't. Okay. 741 01:21:43.890 --> 01:21:48.960 Silence. 742 01:21:48.960 --> 01:21:54.810 Thanks. 743 01:21:54.810 --> 01:22:00.779 Okay, see, if I can. 744 01:22:00.779 --> 01:22:09.810 Silence. 745 01:22:09.810 --> 01:22:20.699 Okay, okay. I don't know if this is legible. 746 01:22:20.699 --> 01:22:24.180 But this was the case of. 747 01:22:26.130 --> 01:22:37.050 The 6 sided guy, and we want the number of ways we could get 11222 3, 3. so we had that multi nomial expression here. 748 01:22:37.050 --> 01:22:41.100 And then we had the probability we divided by 6 to the. 749 01:22:42.539 --> 01:22:48.449 Okay, because it probably of each face coming up with a 10. now if we had the students, so we've got 7. 750 01:22:48.449 --> 01:22:52.229 Outcomes Sunday, Monday, Tuesday, Wednesday, Thursday, Friday, Saturday. 751 01:22:52.229 --> 01:22:56.579 And 21 students on Saturday dividing by 6 to the. 752 01:22:56.579 --> 01:23:02.039 We divide by 21 to the 7th. 753 01:23:02.039 --> 01:23:11.729 If you can read this page that I'm holding up, does that make sense? 754 01:23:11.729 --> 01:23:18.779 Yeah, I think after today's lecture, I'm going to look at it again and Nobel. I'll understand it easier. 755 01:23:18.779 --> 01:23:25.949 The question I have the. 756 01:23:25.949 --> 01:23:36.810 Question 6, well, I can't do 6 both ways because it's ambiguous so, you know, I use it or I don't use it. So. 757 01:23:40.350 --> 01:23:48.420 So, if it's ambiguous, and I make an argument to the grade, or maybe you'll give a bonus point or something. 758 01:23:50.579 --> 01:23:55.079 Other questions sure. 759 01:23:55.079 --> 01:23:58.739 For question yes. What are we reading? 760 01:23:58.739 --> 01:24:04.199 Question 9 yeah. 761 01:24:04.199 --> 01:24:08.819 Anything find some current policy issue that is not controversial or something. 762 01:24:08.819 --> 01:24:13.800 Basic probabilities. I told you think that'd be creative. 763 01:24:13.800 --> 01:24:17.939 Are we going to get a closer later this week? Are you think the. 764 01:24:17.939 --> 01:24:21.840 Whether people are getting their probabilities wrong or something. 765 01:24:23.100 --> 01:24:27.539 All right, so, no, thank you. 766 01:24:27.539 --> 01:24:31.439 Yep, try to tie the course to the real world. 767 01:24:33.869 --> 01:24:37.920 Other questions. 768 01:24:37.920 --> 01:24:41.520 Okay. 769 01:24:42.569 --> 01:24:47.430 In which case that's. 770 01:24:48.510 --> 01:24:56.010 Silence. 771 01:24:56.010 --> 01:24:59.760 Silence. 772 01:24:59.760 --> 01:25:02.819 Okay, so see, you. 773 01:25:02.819 --> 01:25:04.229 Thursday, then.