1 00:00:00,060 --> 00:00:02,490 The following content is, provided under a Creative 2 00:00:02,490 --> 00:00:04,030 Commons license. 3 00:00:04,030 --> 00:00:06,330 Your support will help MIT OpenCourseWare 4 00:00:06,330 --> 00:00:10,690 continue to offer high quality educational resources for free. 5 00:00:10,690 --> 00:00:13,320 To make a donation or view additional materials 6 00:00:13,320 --> 00:00:17,250 from hundreds of MIT courses, visit MIT OpenCourseWare 7 00:00:17,250 --> 00:00:20,164 at ocw.mit.edu 8 00:00:20,164 --> 00:00:20,830 VINA NGUYEN: OK. 9 00:00:20,830 --> 00:00:22,960 So do you guys understand what's going on? 10 00:00:22,960 --> 00:00:23,856 All right. 11 00:00:23,856 --> 00:00:24,355 OK. 12 00:00:24,355 --> 00:00:28,340 So there's bacteria here, and then a duplication process. 13 00:00:28,340 --> 00:00:31,040 They can either duplicate or die. 14 00:00:31,040 --> 00:00:33,060 And that's just for one bacteria. 15 00:00:33,060 --> 00:00:38,031 So you have 2, then that one remains unchanged. 16 00:00:38,031 --> 00:00:39,915 Does that make sense? 17 00:00:39,915 --> 00:00:42,640 OK. 18 00:00:42,640 --> 00:00:46,670 So Question 1 was, what does assumption two 19 00:00:46,670 --> 00:00:49,240 mean, in terms of percentages? 20 00:00:49,240 --> 00:00:52,210 What's the probability that a cell will duplicate, 21 00:00:52,210 --> 00:00:56,780 and what's the probability that the cell will die? 22 00:00:56,780 --> 00:00:58,196 AUDIENCE: [INAUDIBLE] 23 00:00:58,196 --> 00:00:59,612 VINA NGUYEN: Louder. 24 00:00:59,612 --> 00:01:00,556 AUDIENCE: What? 25 00:01:00,556 --> 00:01:01,389 VINA NGUYEN: Louder. 26 00:01:01,389 --> 00:01:02,731 AUDIENCE: It's 50/50. 27 00:01:02,731 --> 00:01:03,480 VINA NGUYEN: Yeah. 28 00:01:03,480 --> 00:01:05,512 So 1/2, right? 29 00:01:05,512 --> 00:01:07,542 So this is one half. 30 00:01:07,542 --> 00:01:10,150 That's all that question is asking. 31 00:01:10,150 --> 00:01:13,545 AUDIENCE: Actually 0. 32 00:01:13,545 --> 00:01:15,000 Or if you're at 4 [INAUDIBLE]. 33 00:01:15,000 --> 00:01:15,708 VINA NGUYEN: Yes. 34 00:01:15,708 --> 00:01:19,030 But this is just for assuming an initial state of 1. 35 00:01:35,200 --> 00:01:36,200 OK. 36 00:01:36,200 --> 00:01:37,290 Anyone for Question 2? 37 00:01:37,290 --> 00:01:39,130 Give me an example of a chain where 38 00:01:39,130 --> 00:01:44,101 the bacteria neither regions n equals 0 or n equals 4. 39 00:01:44,101 --> 00:01:45,642 AUDIENCE: It succeeds, then it fails. 40 00:01:45,642 --> 00:01:46,766 It succeeds, then it fails. 41 00:01:46,766 --> 00:01:48,810 It succeeds, then it fails forever. 42 00:01:48,810 --> 00:01:49,560 VINA NGUYEN: What? 43 00:01:49,560 --> 00:01:51,310 AUDIENCE: It succeeds, then fails forever. 44 00:01:51,310 --> 00:01:53,422 It succeeds, fails, succeeds, fails. 45 00:01:53,422 --> 00:01:54,172 VINA NGUYEN: Yeah. 46 00:01:54,172 --> 00:01:57,632 So an example of a chain would be right. 47 00:02:00,470 --> 00:02:02,835 Yeah. 48 00:02:02,835 --> 00:02:10,995 It can also go from 1, 2, 3, 2, 1, et cetera, et cetera. 49 00:02:10,995 --> 00:02:12,880 OK. 50 00:02:12,880 --> 00:02:13,941 So that's pretty easy. 51 00:02:17,790 --> 00:02:22,070 This path, right here, is called a trajectory in terminology. 52 00:02:22,070 --> 00:02:24,710 And attainable, If you guys read that, 53 00:02:24,710 --> 00:02:26,840 means that you can reach a certain state 54 00:02:26,840 --> 00:02:28,070 from another state. 55 00:02:28,070 --> 00:02:31,430 So we can reach state 3 from state 1. 56 00:02:31,430 --> 00:02:34,310 You can't reach states 3 from state zero. 57 00:02:34,310 --> 00:02:37,250 So that means unattainable. 58 00:02:37,250 --> 00:02:39,820 You can't have bacteria that's non-existential. 59 00:02:45,260 --> 00:02:45,760 OK. 60 00:02:45,760 --> 00:02:51,810 So for question three, did anybody have five generations? 61 00:02:51,810 --> 00:02:52,435 AUDIENCE: Yeah. 62 00:02:55,393 --> 00:03:01,320 2, 3, 2, 1, 0. 63 00:03:01,320 --> 00:03:02,346 VINA NGUYEN: OK. 64 00:03:02,346 --> 00:03:04,667 Anyone for 4? 65 00:03:04,667 --> 00:03:06,000 AUDIENCE: Isn't that impossible? 66 00:03:06,000 --> 00:03:06,750 VINA NGUYEN: Yeah. 67 00:03:06,750 --> 00:03:07,630 So it's not possible. 68 00:03:07,630 --> 00:03:12,360 So you need to make sure you get your problem space correct, 69 00:03:12,360 --> 00:03:15,960 because some paths cannot exist, and some can. 70 00:03:15,960 --> 00:03:19,086 So that's the important part of defining your model. 71 00:03:19,086 --> 00:03:20,460 And if these aren't clear to you, 72 00:03:20,460 --> 00:03:22,798 then you need to redefine the model. 73 00:03:22,798 --> 00:03:25,178 OK? 74 00:03:25,178 --> 00:03:29,000 All right, so Question 4. 75 00:03:29,000 --> 00:03:30,430 I know a lot of you guys asked. 76 00:03:30,430 --> 00:03:31,850 It's my fault. Transition probabilities 77 00:03:31,850 --> 00:03:33,266 are the probabilities that you can 78 00:03:33,266 --> 00:03:36,380 reach a state to another state in just one step. 79 00:03:38,891 --> 00:03:39,390 Yeah. 80 00:03:39,390 --> 00:03:41,760 So if I give you-- 81 00:03:41,760 --> 00:03:44,060 this is a transition probability. 82 00:03:44,060 --> 00:03:45,890 So this is telling you the probability 83 00:03:45,890 --> 00:03:53,140 that you go from state I to state J. And I or J can-- 84 00:03:53,140 --> 00:03:55,760 which is what n was-- 85 00:03:55,760 --> 00:04:02,870 can be either state 0, 1, 2, 3, or 4. 86 00:04:02,870 --> 00:04:05,360 So what I am asking you is the probability that you go from 87 00:04:05,360 --> 00:04:09,890 state 0 to state 0 is-- or the probability that you go from 88 00:04:09,890 --> 00:04:16,492 state 1 to 0, or the probability that you go from 1 to 2-- 89 00:04:16,492 --> 00:04:17,971 something like that. 90 00:04:23,410 --> 00:04:25,664 So for these transition probabilities, 91 00:04:25,664 --> 00:04:27,656 can anyone tell me what this is? 92 00:04:27,656 --> 00:04:28,160 AUDIENCE: 1. 93 00:04:28,160 --> 00:04:28,740 VINA NGUYEN: Yeah. 94 00:04:28,740 --> 00:04:30,060 AUDIENCE: Or we could just look at this. 95 00:04:30,060 --> 00:04:31,440 VINA NGUYEN: Yeah, I know. 96 00:04:31,440 --> 00:04:32,160 This one? 97 00:04:35,648 --> 00:04:37,356 AUDIENCE: Yeah, I can't read the numbers. 98 00:04:37,356 --> 00:04:37,730 VINA NGUYEN: Oh. 99 00:04:37,730 --> 00:04:38,938 AUDIENCE: Can't see that far. 100 00:04:38,938 --> 00:04:42,350 VINA NGUYEN: 1, 0, 1/2. 101 00:04:42,350 --> 00:04:43,680 Yeah. 102 00:04:43,680 --> 00:04:44,930 1 to 2 is 1/2, right? 103 00:04:44,930 --> 00:04:48,000 And then 4 to 4 is unrealistic, but for is problem 104 00:04:48,000 --> 00:04:49,395 it's going to be 1. 105 00:04:49,395 --> 00:04:50,790 OK? 106 00:04:50,790 --> 00:04:52,370 So that's what it's asking you. 107 00:04:55,770 --> 00:04:58,410 Question 5 is more of a [? thought ?] answer. 108 00:04:58,410 --> 00:05:03,980 So did you guys read beforehand the concepts, 109 00:05:03,980 --> 00:05:05,430 how that affects it? 110 00:05:05,430 --> 00:05:05,930 OK. 111 00:05:05,930 --> 00:05:07,471 AUDIENCE: Wait, what's the difference 112 00:05:07,471 --> 00:05:08,612 between the two concepts? 113 00:05:08,612 --> 00:05:09,403 VINA NGUYEN: Right. 114 00:05:09,403 --> 00:05:10,490 [INAUDIBLE] OK. 115 00:05:10,490 --> 00:05:16,610 So the first one is referring to not the actual states, 116 00:05:16,610 --> 00:05:19,930 but more like your actual problem-- 117 00:05:19,930 --> 00:05:23,140 so how the bacteria actually changes, 118 00:05:23,140 --> 00:05:24,649 if that affects your problem. 119 00:05:24,649 --> 00:05:25,190 AUDIENCE: Oh. 120 00:05:25,190 --> 00:05:26,815 VINA NGUYEN: But the second one is just 121 00:05:26,815 --> 00:05:40,560 based on the number, which is [? your state. ?] 122 00:05:40,560 --> 00:05:45,130 So for this model to work, if you're at state 3, 123 00:05:45,130 --> 00:05:48,100 then it doesn't matter what states you're at before 3. 124 00:05:48,100 --> 00:05:51,141 What only matters is that you're at state 3 right now. 125 00:05:51,141 --> 00:05:51,640 OK. 126 00:05:51,640 --> 00:05:52,650 So does that make sense? 127 00:05:52,650 --> 00:05:53,954 AUDIENCE: Yeah. 128 00:05:53,954 --> 00:05:54,620 VINA NGUYEN: OK. 129 00:05:54,620 --> 00:05:56,955 AUDIENCE: So that would be number two. 130 00:05:56,955 --> 00:05:57,890 VINA NGUYEN: Mm-hmm. 131 00:05:57,890 --> 00:06:00,849 And that's called the Markov property. 132 00:06:08,833 --> 00:06:09,340 OK. 133 00:06:09,340 --> 00:06:12,943 So did you guys figure out 6 at all? 134 00:06:12,943 --> 00:06:14,270 AUDIENCE: What was Question 6? 135 00:06:14,270 --> 00:06:16,630 VINA NGUYEN: So transition probabilities 136 00:06:16,630 --> 00:06:19,980 are going from one state to the next in one step, right? 137 00:06:19,980 --> 00:06:23,170 But now I'm asking, what are the probabilities? 138 00:06:23,170 --> 00:06:24,021 Two steps. 139 00:06:33,160 --> 00:06:35,310 So I would do 6, 7, and 8 at once, 140 00:06:35,310 --> 00:06:38,735 then, because those answer better using matrices. 141 00:06:48,090 --> 00:06:51,040 You guys have this diagram already, right? 142 00:06:51,040 --> 00:06:53,030 It's not labelled, but it's the fourth figure. 143 00:07:17,440 --> 00:07:18,810 You guys have this, right? 144 00:07:18,810 --> 00:07:21,390 OK, so this is just for one step. 145 00:07:21,390 --> 00:07:25,005 But did anyone figure out how to get this matrix for two steps? 146 00:07:28,965 --> 00:07:30,450 Yeah. 147 00:07:30,450 --> 00:07:33,750 AUDIENCE: [INAUDIBLE] you got the same thing that I got 148 00:07:33,750 --> 00:07:35,827 when I actually did it at home. 149 00:07:35,827 --> 00:07:37,785 VINA NGUYEN: So you mean multiply it by itself? 150 00:07:37,785 --> 00:07:38,575 AUDIENCE: Yeah. 151 00:07:38,575 --> 00:07:39,325 VINA NGUYEN: Yeah. 152 00:07:39,325 --> 00:07:42,830 So that would get you the second. 153 00:07:42,830 --> 00:07:46,471 So I'll just do that [INAUDIBLE].. 154 00:07:46,471 --> 00:07:47,346 AUDIENCE: [INAUDIBLE] 155 00:07:47,346 --> 00:07:48,813 VINA NGUYEN: No, I'm OK. 156 00:07:55,659 --> 00:07:57,126 AUDIENCE: Oh, it's an eraser. 157 00:07:57,126 --> 00:08:01,360 VINA NGUYEN: You got an eraser over there? 158 00:08:01,360 --> 00:08:01,860 OK. 159 00:08:01,860 --> 00:08:02,772 I'll just do it here. 160 00:08:02,772 --> 00:08:03,271 OK. 161 00:08:03,271 --> 00:08:05,960 So you guys-- hm? 162 00:08:05,960 --> 00:08:09,340 AUDIENCE: There's one right there, behind it. 163 00:08:09,340 --> 00:08:11,830 VINA NGUYEN: In this one? 164 00:08:11,830 --> 00:08:13,324 It's just-- 165 00:08:13,324 --> 00:08:15,314 AUDIENCE: No, no. 166 00:08:15,314 --> 00:08:15,814 Yeah. 167 00:08:15,814 --> 00:08:18,304 Push that one, the one you're holding. 168 00:08:23,284 --> 00:08:25,587 VINA NGUYEN: Yeah, I used to have three. 169 00:08:25,587 --> 00:08:26,578 Oh, here it is. 170 00:08:26,578 --> 00:08:27,078 OK. 171 00:08:33,539 --> 00:08:36,414 So quick review if you guys don't know how 172 00:08:36,414 --> 00:08:39,396 to do matrix multiplication. 173 00:08:39,396 --> 00:08:41,020 You already know? 174 00:08:41,020 --> 00:08:41,610 AUDIENCE: No. 175 00:08:41,610 --> 00:08:42,380 VINA NGUYEN: No? 176 00:08:42,380 --> 00:08:46,132 OK, I'll just go over it. 177 00:08:46,132 --> 00:08:49,270 OK, so for you guys who do know, what's this? 178 00:08:59,240 --> 00:09:01,610 AUDIENCE: ab equals [INAUDIBLE]. 179 00:09:04,444 --> 00:09:06,799 VINA NGUYEN: Right? 180 00:09:06,799 --> 00:09:09,625 Does that make sense? 181 00:09:09,625 --> 00:09:14,236 This row, this column, this row, this column. 182 00:09:14,236 --> 00:09:14,735 Right? 183 00:09:14,735 --> 00:09:22,751 So aa-- aa, bc, ab, dd. 184 00:09:22,751 --> 00:09:23,251 Right? 185 00:09:23,251 --> 00:09:23,751 OK. 186 00:09:23,751 --> 00:09:24,712 So which one is this? 187 00:09:24,712 --> 00:09:33,965 AUDIENCE: [INAUDIBLE] 188 00:09:33,965 --> 00:09:34,939 VINA NGUYEN: d. 189 00:09:34,939 --> 00:09:36,400 OK. 190 00:09:36,400 --> 00:09:38,840 Everyone get that? 191 00:09:38,840 --> 00:09:40,610 You'll get it more as you see numbers. 192 00:09:40,610 --> 00:09:41,110 OK. 193 00:09:44,310 --> 00:09:51,540 So we have, for our case , b00, b11, et cetera, et cetera. 194 00:09:51,540 --> 00:09:59,900 So if you multiply it out together, then what do you get? 195 00:09:59,900 --> 00:10:06,720 You get p00 times p00, right? 196 00:10:12,696 --> 00:10:15,684 Can anyone tell me that? 197 00:10:15,684 --> 00:10:16,680 What's next? 198 00:10:24,646 --> 00:10:25,146 Right? 199 00:10:39,588 --> 00:10:42,576 Yeah, I'm just going to do the first two-- 200 00:10:42,576 --> 00:10:44,444 the first two entries. 201 00:10:51,080 --> 00:10:54,380 So you can just multiply it, but you need 202 00:10:54,380 --> 00:10:55,930 to understand why that works. 203 00:10:55,930 --> 00:10:59,240 So this tells you how it works, because it goes from 0 to 0, 204 00:10:59,240 --> 00:11:03,170 0 back to 0, which is what you want in two steps. 205 00:11:03,170 --> 00:11:06,140 And then this is 0 to 1, and back to 1 to 0, 206 00:11:06,140 --> 00:11:07,940 just two steps again. 207 00:11:07,940 --> 00:11:08,930 Right? 208 00:11:08,930 --> 00:11:10,560 Same thing for two-- 209 00:11:10,560 --> 00:11:12,802 0 to 2, 2 to 0, et cetera. 210 00:11:16,658 --> 00:11:18,710 And the second entry the same way, 211 00:11:18,710 --> 00:11:20,930 so that you have a different initial state. 212 00:11:20,930 --> 00:11:22,890 So 1 to 0, 0 to1. 213 00:11:22,890 --> 00:11:25,645 Or wait-- 00. 214 00:11:29,530 --> 00:11:32,360 Does that make sense to everybody? 215 00:11:32,360 --> 00:11:32,860 OK. 216 00:11:32,860 --> 00:11:34,570 So if you just follow that same formula, 217 00:11:34,570 --> 00:11:37,540 you'll get it for every initial state and every end state 218 00:11:37,540 --> 00:11:39,160 in two steps. 219 00:11:39,160 --> 00:11:42,858 And the easy way to do that is to just multiply the matrix-- 220 00:11:42,858 --> 00:11:44,734 so what you see. 221 00:11:47,548 --> 00:11:52,060 So pn means this transition probably 222 00:11:52,060 --> 00:11:54,910 matrix for one step times the number of steps. 223 00:11:54,910 --> 00:11:58,020 So to get the number two steps, you would just times 224 00:11:58,020 --> 00:12:00,240 it by itself. 225 00:12:00,240 --> 00:12:02,280 Three steps, you just times it by itself. 226 00:12:02,280 --> 00:12:10,874 [INAUDIBLE] So does anyone have a calculator? 227 00:12:10,874 --> 00:12:11,846 Do you? 228 00:12:11,846 --> 00:12:12,818 OK. 229 00:12:12,818 --> 00:12:15,734 AUDIENCE: [INAUDIBLE] matrix with two steps? 230 00:12:15,734 --> 00:12:18,487 VINA NGUYEN: OK. 231 00:12:18,487 --> 00:12:20,070 The question I want to answer, though, 232 00:12:20,070 --> 00:12:25,320 is if any of those values converge if n is large. 233 00:12:25,320 --> 00:12:28,850 So if you multiply your matrix a million times, 234 00:12:28,850 --> 00:12:31,430 does it converge? 235 00:12:31,430 --> 00:12:32,656 Do these numbers converge? 236 00:12:32,656 --> 00:12:33,974 AUDIENCE: [INAUDIBLE] 237 00:12:33,974 --> 00:12:34,640 VINA NGUYEN: Hm? 238 00:12:34,640 --> 00:12:35,600 AUDIENCE: [INAUDIBLE] 239 00:12:35,600 --> 00:12:36,266 VINA NGUYEN: OK. 240 00:12:38,850 --> 00:12:43,305 AUDIENCE: Well, I think it will eventually go to either 0 or 4. 241 00:12:43,305 --> 00:12:46,275 Well, I have the 5 by 5. 242 00:12:46,275 --> 00:12:47,265 [INAUDIBLE] 243 00:12:47,265 --> 00:12:49,245 The probability would be the 3, 2 244 00:12:49,245 --> 00:12:52,215 and you'd have the trajectory go on forever. 245 00:12:52,215 --> 00:12:54,690 [INAUDIBLE] that should be 0. 246 00:12:54,690 --> 00:12:59,640 And then all you'd have to do is [INAUDIBLE] 247 00:12:59,640 --> 00:13:04,095 you should have 3/4 and 1. 248 00:13:04,095 --> 00:13:06,580 [INAUDIBLE] 249 00:13:06,580 --> 00:13:08,370 VINA NGUYEN: So you're getting there. 250 00:13:08,370 --> 00:13:12,580 The matrix will tell you for every single possible step-- 251 00:13:12,580 --> 00:13:15,179 every initial state, every end state. 252 00:13:15,179 --> 00:13:15,970 So if I asked you-- 253 00:13:15,970 --> 00:13:17,350 AUDIENCE: [INAUDIBLE] 254 00:13:17,350 --> 00:13:19,350 VINA NGUYEN: Well if I asked you the probability 255 00:13:19,350 --> 00:13:24,590 that you can go from state two to four with n-- 256 00:13:24,590 --> 00:13:27,790 or two to three with n equals forever, 257 00:13:27,790 --> 00:13:30,262 then that's a little harder, right? 258 00:13:30,262 --> 00:13:37,370 So then you have to multiply all the different combinations. 259 00:13:37,370 --> 00:13:41,070 AUDIENCE: [INAUDIBLE] say, hey, this 260 00:13:41,070 --> 00:13:45,942 is the thing that failed completely in this number set 261 00:13:45,942 --> 00:13:52,380 [INAUDIBLE] 262 00:13:52,380 --> 00:13:54,380 VINA NGUYEN: You know, you might be right. 263 00:13:54,380 --> 00:13:58,160 And I would have to read more, but for now, we're 264 00:13:58,160 --> 00:14:00,182 just doing it this way. 265 00:14:00,182 --> 00:14:03,915 But I can get back to you. 266 00:14:03,915 --> 00:14:08,785 AUDIENCE: [INAUDIBLE] 267 00:14:08,785 --> 00:14:10,246 VINA NGUYEN: Did you get it yet? 268 00:14:10,246 --> 00:14:12,510 AUDIENCE: Well, how many times do you want me to [INAUDIBLE]?? 269 00:14:12,510 --> 00:14:14,343 VINA NGUYEN: Just until the values converge. 270 00:14:14,343 --> 00:14:15,618 AUDIENCE: Until they converge? 271 00:14:15,618 --> 00:14:18,360 [INAUDIBLE] 272 00:14:18,360 --> 00:14:22,532 VINA NGUYEN: Because n is relative. 273 00:14:22,532 --> 00:14:24,760 AUDIENCE: Do I need to know how many times? 274 00:14:24,760 --> 00:14:26,617 VINA NGUYEN: Well, it would be nice. 275 00:14:26,617 --> 00:14:28,090 AUDIENCE: Whoops. 276 00:14:28,090 --> 00:14:29,500 VINA NGUYEN: It's OK. 277 00:14:29,500 --> 00:14:30,850 AUDIENCE: [INAUDIBLE] 278 00:14:30,850 --> 00:14:31,910 VINA NGUYEN: That's fine. 279 00:14:31,910 --> 00:14:35,090 AUDIENCE: I'm just pressing the enter button and seeing-- 280 00:14:35,090 --> 00:14:35,965 VINA NGUYEN: It's OK. 281 00:14:49,120 --> 00:14:51,790 Converge means the values stop chaining. 282 00:14:51,790 --> 00:14:53,510 AUDIENCE: OK, what is values? 283 00:14:53,510 --> 00:14:56,760 VINA NGUYEN: Any-- all the values in the matrix. 284 00:14:56,760 --> 00:14:58,620 AUDIENCE: The first column stopped. 285 00:14:58,620 --> 00:14:59,420 VINA NGUYEN: Right. 286 00:14:59,420 --> 00:15:00,211 How about the rest? 287 00:15:00,211 --> 00:15:01,231 AUDIENCE: Not yet. 288 00:15:01,231 --> 00:15:02,106 VINA NGUYEN: Not yet? 289 00:15:02,106 --> 00:15:03,180 OK. 290 00:15:03,180 --> 00:15:05,910 And you can do it to two decimal points. 291 00:15:05,910 --> 00:15:06,884 That's fine. 292 00:15:11,672 --> 00:15:13,630 AUDIENCE: What's weird is in the second column, 293 00:15:13,630 --> 00:15:16,385 the numbers get fluctuated. 294 00:15:16,385 --> 00:15:17,510 VINA NGUYEN: Second column? 295 00:15:17,510 --> 00:15:20,420 AUDIENCE: Yeah. 296 00:15:20,420 --> 00:15:22,935 VINA NGUYEN: That means they don't converge. 297 00:15:22,935 --> 00:15:23,560 AUDIENCE: Cool. 298 00:15:23,560 --> 00:15:25,120 So yeah, they don't converge. 299 00:15:25,120 --> 00:15:28,270 But the first column-- wait, let me check the third column. 300 00:15:28,270 --> 00:15:28,770 Oh, wait. 301 00:15:28,770 --> 00:15:29,960 No, they do. 302 00:15:29,960 --> 00:15:31,200 It's just getting close to 0. 303 00:15:31,200 --> 00:15:33,522 I didn't see the negative 0.9. 304 00:15:33,522 --> 00:15:34,188 VINA NGUYEN: OK. 305 00:15:34,188 --> 00:15:35,682 Then they do. 306 00:15:35,682 --> 00:15:37,176 AUDIENCE: Yeah, sorry. 307 00:15:37,176 --> 00:15:38,958 VINA NGUYEN: No, it's fine. 308 00:15:38,958 --> 00:15:39,666 AUDIENCE: Whoops. 309 00:15:42,670 --> 00:15:45,035 VINA NGUYEN: Oh, you're right, because it's e to the-- 310 00:15:45,035 --> 00:15:45,660 AUDIENCE: Yeah. 311 00:15:45,660 --> 00:15:47,160 I didn't see that [INAUDIBLE]. 312 00:15:47,160 --> 00:15:49,158 AUDIENCE: Hey, look at the [INAUDIBLE].. 313 00:15:49,158 --> 00:15:49,824 VINA NGUYEN: OK. 314 00:15:49,824 --> 00:15:50,873 So you do have a value? 315 00:15:50,873 --> 00:15:51,498 AUDIENCE: Yeah. 316 00:15:51,498 --> 00:15:52,164 VINA NGUYEN: OK. 317 00:15:52,164 --> 00:15:53,933 So what's your matrix? 318 00:15:53,933 --> 00:15:55,766 AUDIENCE: OK, this is going to take a while. 319 00:15:55,766 --> 00:15:57,092 [INAUDIBLE] 320 00:15:57,092 --> 00:16:02,954 OK, 1, 0, 0, 0-- 321 00:16:02,954 --> 00:16:04,370 VINA NGUYEN: You can go by column. 322 00:16:04,370 --> 00:16:04,911 AUDIENCE: Oh. 323 00:16:04,911 --> 00:16:06,323 VINA NGUYEN: It might be-- 324 00:16:06,323 --> 00:16:09,009 yeah. 325 00:16:09,009 --> 00:16:09,550 AUDIENCE: OK. 326 00:16:09,550 --> 00:16:15,520 1.75, 0.5, 0.25, and 0. 327 00:16:15,520 --> 00:16:24,394 And then 0-- these are basically 0s all the way down. 328 00:16:24,394 --> 00:16:27,710 And then 0s all the way down. 329 00:16:27,710 --> 00:16:28,664 Well, 0, 0, 1, 0, 0. 330 00:16:28,664 --> 00:16:29,580 VINA NGUYEN: This one? 331 00:16:29,580 --> 00:16:31,090 AUDIENCE: Yeah, I think so. 332 00:16:31,090 --> 00:16:31,590 Wait. 333 00:16:31,590 --> 00:16:32,910 VINA NGUYEN: No, this is 0. 334 00:16:32,910 --> 00:16:33,990 AUDIENCE: That's new. 335 00:16:33,990 --> 00:16:37,853 And then all 0s again. 336 00:16:37,853 --> 00:16:40,850 And the last one's just the first one reversed. 337 00:16:40,850 --> 00:16:47,512 So 0, 0.25, 0.5, 0.75, 1. 338 00:16:47,512 --> 00:16:48,470 VINA NGUYEN: All right. 339 00:16:48,470 --> 00:16:51,960 So does everyone see what this is saying? 340 00:16:51,960 --> 00:16:54,832 Basically, if you repeat this experiment for n 341 00:16:54,832 --> 00:16:59,190 equals forever, or some really large number, then you 342 00:16:59,190 --> 00:17:03,060 can figure out a probability for this entire model. 343 00:17:03,060 --> 00:17:14,660 Then you can go from state 0 to 0, state 1 to 0, state 4 to 4, 344 00:17:14,660 --> 00:17:16,995 or state 3 to 4, et cetera. 345 00:17:24,430 --> 00:17:27,280 So from your initial problem, you only 346 00:17:27,280 --> 00:17:28,740 know that it's half, half, half. 347 00:17:28,740 --> 00:17:31,360 And you have no idea, really, how the experiment is going 348 00:17:31,360 --> 00:17:33,240 to end up in the long run. 349 00:17:33,240 --> 00:17:36,280 But if you do this thing you can see, oh, 350 00:17:36,280 --> 00:17:38,260 well if I start off with three, then 351 00:17:38,260 --> 00:17:40,280 I have a greater chance ending on 4. 352 00:17:40,280 --> 00:17:43,760 [INAUDIBLE] a 75% chance of getting to n. 353 00:17:43,760 --> 00:17:46,930 But if I start, like, initial state 1, 354 00:17:46,930 --> 00:17:52,520 the I have, in the long run, 75% chance of dying out. 355 00:17:52,520 --> 00:17:54,860 So this tells you all the possibilities 356 00:17:54,860 --> 00:17:58,581 that can happen if you run it over and over and over. 357 00:17:58,581 --> 00:18:00,822 Does that make sense? 358 00:18:00,822 --> 00:18:03,280 So the important thing to notice is that your initial state 359 00:18:03,280 --> 00:18:08,250 does matter in how it ends up, even if the probabilities seem 360 00:18:08,250 --> 00:18:10,074 pretty much the same initially. 361 00:18:20,508 --> 00:18:21,008 OK. 362 00:18:21,008 --> 00:18:24,770 So that's Question 9, I thin. 363 00:18:24,770 --> 00:18:26,750 Yeah. 364 00:18:26,750 --> 00:18:31,360 So a Markov model is illustrated usually in this way. 365 00:18:31,360 --> 00:18:32,910 So you have the states like this. 366 00:18:38,630 --> 00:18:41,405 And then you'll draw arrows that represent the probabilities. 367 00:18:41,405 --> 00:18:46,492 So 0 goes back to 0 probability of 1. 368 00:18:46,492 --> 00:18:50,300 And then 1, 2, 0 is 1/2. 369 00:18:50,300 --> 00:18:52,363 1 to 2 is 1/2, et cetera. 370 00:18:55,660 --> 00:19:00,515 So this is a new way of describing your problem space, 371 00:19:00,515 --> 00:19:01,015 right? 372 00:19:01,015 --> 00:19:04,480 This is what's going on in your entire model. 373 00:19:04,480 --> 00:19:07,380 Because unlike before where we had specific outcomes-- 374 00:19:07,380 --> 00:19:09,570 you could figure out the probability of each-- 375 00:19:09,570 --> 00:19:12,690 this one keeps alternating. 376 00:19:12,690 --> 00:19:16,952 So that's basically what it's trying to tell you. 377 00:19:31,704 --> 00:19:32,204 OK? 378 00:19:41,580 --> 00:19:43,390 And if you guys did read it, right here 379 00:19:43,390 --> 00:19:47,030 I think it tells you that state 0 and 4 are called absorbing 380 00:19:47,030 --> 00:19:51,620 states because that's where ultimately your experiment will 381 00:19:51,620 --> 00:19:52,255 end up. 382 00:19:52,255 --> 00:19:57,540 So they're like endpoints for this model. 383 00:19:57,540 --> 00:19:58,040 OK? 384 00:19:58,040 --> 00:20:00,700 So does that make sense, everybody? 385 00:20:00,700 --> 00:20:04,250 So Number 1, did anyone get an answer? 386 00:20:04,250 --> 00:20:05,805 AUDIENCE: [INAUDIBLE] 387 00:20:05,805 --> 00:20:06,555 VINA NGUYEN: Yeah? 388 00:20:06,555 --> 00:20:07,680 AUDIENCE: 10%? 389 00:20:07,680 --> 00:20:08,430 VINA NGUYEN: Yeah. 390 00:20:08,430 --> 00:20:09,730 Did everyone get that? 391 00:20:09,730 --> 00:20:10,660 AUDIENCE: Yeah. 392 00:20:10,660 --> 00:20:12,234 VINA NGUYEN: Everyone understand it? 393 00:20:12,234 --> 00:20:12,915 AUDIENCE: Yeah. 394 00:20:12,915 --> 00:20:14,330 VINA NGUYEN: Anyone want me to go over it? 395 00:20:14,330 --> 00:20:14,926 AUDIENCE: No. 396 00:20:14,926 --> 00:20:15,700 VINA NGUYEN: OK. 397 00:20:15,700 --> 00:20:17,025 So that's an easy-- 398 00:20:17,025 --> 00:20:19,100 OK. 399 00:20:19,100 --> 00:20:19,600 2? 400 00:20:22,790 --> 00:20:25,378 AUDIENCE: [INAUDIBLE] 401 00:20:25,378 --> 00:20:27,306 VINA NGUYEN: Can you give me in fractions? 402 00:20:27,306 --> 00:20:31,162 AUDIENCE: 95 times 94 times 96 times 92 over 100 times 403 00:20:31,162 --> 00:20:34,070 99 times 98 times 97. 404 00:20:34,070 --> 00:20:36,400 VINA NGUYEN: Did everyone get that? 405 00:20:36,400 --> 00:20:43,190 So-- right? 406 00:20:43,190 --> 00:20:47,390 Because there is 95 [INAUDIBLE]. 407 00:20:47,390 --> 00:20:50,460 And then every time you choose one, you have to take one out. 408 00:20:53,330 --> 00:20:53,830 OK? 409 00:20:53,830 --> 00:20:56,460 Does that make sense to everybody? 410 00:20:56,460 --> 00:20:57,161 So is that hard? 411 00:20:57,161 --> 00:20:57,660 Easy? 412 00:20:57,660 --> 00:20:58,440 Medium? 413 00:20:58,440 --> 00:20:59,064 AUDIENCE: Wait. 414 00:20:59,064 --> 00:21:00,124 Shouldn't there be one-- 415 00:21:00,124 --> 00:21:01,582 oh, but that was just [INAUDIBLE].. 416 00:21:01,582 --> 00:21:02,082 Never mind. 417 00:21:02,082 --> 00:21:03,990 I counted wrong. 418 00:21:03,990 --> 00:21:06,887 VINA NGUYEN: Is this harder than the last one? 419 00:21:06,887 --> 00:21:07,470 AUDIENCE: Yep. 420 00:21:07,470 --> 00:21:08,678 VINA NGUYEN: A little harder? 421 00:21:08,678 --> 00:21:10,847 AUDIENCE: Just that one was easy. 422 00:21:10,847 --> 00:21:11,930 VINA NGUYEN: Just curious. 423 00:21:15,020 --> 00:21:16,020 3? 424 00:21:16,020 --> 00:21:17,762 Anyone got an answer? 425 00:21:17,762 --> 00:21:18,345 AUDIENCE: 2/3? 426 00:21:18,345 --> 00:21:19,095 VINA NGUYEN: Yeah. 427 00:21:19,095 --> 00:21:20,880 Did you get that too? 428 00:21:20,880 --> 00:21:21,550 OK. 429 00:21:21,550 --> 00:21:23,250 Well, I'll go over it, then. 430 00:21:23,250 --> 00:21:26,770 All right, so there's four combinations of kids, right? 431 00:21:35,633 --> 00:21:36,495 You get it now? 432 00:21:36,495 --> 00:21:37,360 OK. 433 00:21:37,360 --> 00:21:41,510 So if they have a boy, it can be this. 434 00:21:41,510 --> 00:21:46,160 And there's 2/3 chance that he has it. 435 00:21:46,160 --> 00:21:49,660 AUDIENCE: [INAUDIBLE] 2/3 [INAUDIBLE] has the same thing? 436 00:21:49,660 --> 00:21:52,160 Like, [INAUDIBLE]? 437 00:21:52,160 --> 00:21:53,160 VINA NGUYEN: Oh, OK. 438 00:21:53,160 --> 00:21:53,659 Yeah. 439 00:21:53,659 --> 00:21:55,830 So it's different, yeah-- 440 00:21:55,830 --> 00:21:58,822 because it matters which order he can-- 441 00:21:58,822 --> 00:22:00,806 AUDIENCE: Well, it's still really confusing 442 00:22:00,806 --> 00:22:03,440 because it's about meeting the boys. 443 00:22:03,440 --> 00:22:06,700 In the third example, you wouldn't have met the boy. 444 00:22:06,700 --> 00:22:07,906 AUDIENCE: Why not? 445 00:22:07,906 --> 00:22:10,236 Well, they're both boys, and they're both alive. 446 00:22:10,236 --> 00:22:11,610 So you could have met either one. 447 00:22:11,610 --> 00:22:13,401 Doesn't matter if they're older or younger. 448 00:22:13,401 --> 00:22:15,260 AUDIENCE: Yes, but with those two, 449 00:22:15,260 --> 00:22:17,851 then you have the same chance of meeting a girl. 450 00:22:17,851 --> 00:22:19,960 Like, with both of them, you have 3% chance that-- 451 00:22:19,960 --> 00:22:20,740 VINA NGUYEN: But there's no chance 452 00:22:20,740 --> 00:22:22,720 whether you meet either one, because I'm telling you 453 00:22:22,720 --> 00:22:23,460 you are meeting a boy. 454 00:22:23,460 --> 00:22:23,970 AUDIENCE: Right. 455 00:22:23,970 --> 00:22:24,720 VINA NGUYEN: Yeah. 456 00:22:24,720 --> 00:22:26,332 So it's an assumption, right? 457 00:22:26,332 --> 00:22:27,248 AUDIENCE: [INAUDIBLE]. 458 00:22:27,248 --> 00:22:28,830 I don't know. 459 00:22:28,830 --> 00:22:30,496 AUDIENCE: Is someone's cell phone going, 460 00:22:30,496 --> 00:22:32,010 or is that just another room? 461 00:22:32,010 --> 00:22:33,630 AUDIENCE: No, that's the other room. 462 00:22:33,630 --> 00:22:35,684 They're playing Cindy Lauper. 463 00:22:35,684 --> 00:22:37,160 AUDIENCE: Where is this other room? 464 00:22:37,160 --> 00:22:38,160 VINA NGUYEN: That's odd. 465 00:22:41,500 --> 00:22:42,000 4? 466 00:22:45,840 --> 00:22:46,837 AUDIENCE: 45? 467 00:22:46,837 --> 00:22:47,420 AUDIENCE: 45%. 468 00:22:47,420 --> 00:22:48,240 0.45. 469 00:22:48,240 --> 00:22:49,865 VINA NGUYEN: Oh, I didn't calculate it, 470 00:22:49,865 --> 00:22:51,421 so I thought you did the-- 471 00:22:51,421 --> 00:22:53,050 the way to do it. 472 00:22:53,050 --> 00:22:54,140 AUDIENCE: 0.3 times 0.5. 473 00:22:54,140 --> 00:22:56,140 VINA NGUYEN: OK. 474 00:22:56,140 --> 00:23:04,033 AUDIENCE: Plus 0.4 times 0.25 plus 0.8 times 0.5. 475 00:23:04,033 --> 00:23:05,241 VINA NGUYEN: So what is that? 476 00:23:05,241 --> 00:23:05,949 What did you say? 477 00:23:05,949 --> 00:23:06,690 AUDIENCE: 0.45. 478 00:23:06,690 --> 00:23:07,615 VINA NGUYEN: OK. 479 00:23:07,615 --> 00:23:08,115 All right. 480 00:23:08,115 --> 00:23:10,281 So you shouldn't play, because you'll lose. 481 00:23:14,540 --> 00:23:15,183 Yeah? 482 00:23:15,183 --> 00:23:17,307 AUDIENCE: 8 times 70,000. 483 00:23:17,307 --> 00:23:18,265 VINA NGUYEN: Oh, sorry. 484 00:23:18,265 --> 00:23:21,590 I-- OK, yeah. 485 00:23:21,590 --> 00:23:22,798 AUDIENCE: No, it's 8 million. 486 00:23:22,798 --> 00:23:24,381 VINA NGUYEN: Everyone understand this? 487 00:23:24,381 --> 00:23:27,422 So you only have 8 options for your first digit. 488 00:23:27,422 --> 00:23:33,194 And then for the rest, you have 10 options. 489 00:23:33,194 --> 00:23:34,156 AUDIENCE: Oh, congrats. 490 00:23:34,156 --> 00:23:34,656 0. 491 00:23:34,656 --> 00:23:37,616 Yay! 492 00:23:37,616 --> 00:23:39,600 VINA NGUYEN: Does that make sense? 493 00:23:39,600 --> 00:23:40,394 I forgot to ask. 494 00:23:40,394 --> 00:23:44,432 So 3-- easy, medium, hard? 495 00:23:44,432 --> 00:23:45,842 Medium? 496 00:23:45,842 --> 00:23:46,550 AUDIENCE: Medium. 497 00:23:46,550 --> 00:23:47,510 Yeah, medium. 498 00:23:47,510 --> 00:23:48,680 Hard. 499 00:23:48,680 --> 00:23:49,430 VINA NGUYEN: Hard. 500 00:23:49,430 --> 00:23:49,930 OK. 501 00:23:49,930 --> 00:23:51,410 How about 4? 502 00:23:51,410 --> 00:23:53,220 AUDIENCE: Easy. 503 00:23:53,220 --> 00:23:54,444 VINA NGUYEN: 5? 504 00:23:54,444 --> 00:23:56,610 AUDIENCE: Easy. 505 00:23:56,610 --> 00:23:59,237 VINA NGUYEN: So for 6, did anyone get that? 506 00:23:59,237 --> 00:24:00,400 AUDIENCE: Hard. 507 00:24:00,400 --> 00:24:03,450 VINA NGUYEN: So pmf is basically the probability 508 00:24:03,450 --> 00:24:05,950 that your random variable can be [INAUDIBLE].. 509 00:24:05,950 --> 00:24:08,450 And I'll kind of explain that. 510 00:24:08,450 --> 00:24:12,170 And I'll upload the file so you can look at it. 511 00:24:12,170 --> 00:24:14,952 Did anyone get 6? 512 00:24:14,952 --> 00:24:20,940 AUDIENCE: [INAUDIBLE] 513 00:24:20,940 --> 00:24:23,590 VINA NGUYEN: Probability math's not good. 514 00:24:23,590 --> 00:24:30,530 OK, so this is-- 515 00:24:30,530 --> 00:24:31,700 AUDIENCE: This is hard? 516 00:24:31,700 --> 00:24:32,783 VINA NGUYEN: This is hard. 517 00:24:32,783 --> 00:24:33,790 OK. 518 00:24:33,790 --> 00:24:35,280 I see. 519 00:24:35,280 --> 00:24:37,460 Do you remember this one? 520 00:24:37,460 --> 00:24:40,180 I know you have. 521 00:24:40,180 --> 00:24:42,846 AUDIENCE: Oh. 522 00:24:42,846 --> 00:24:45,286 Kind of, yeah. 523 00:24:45,286 --> 00:24:52,118 AUDIENCE: [INAUDIBLE] 524 00:24:52,118 --> 00:24:56,026 VINA NGUYEN: This is binomial, right? 525 00:24:56,026 --> 00:24:56,902 OK, yeah. 526 00:24:56,902 --> 00:24:58,216 OK, good. 527 00:24:58,216 --> 00:25:02,242 So to answer your question, since you weren't here, 528 00:25:02,242 --> 00:25:06,510 k is the different numbers that your [? random interval ?] 529 00:25:06,510 --> 00:25:07,590 can take, right? 530 00:25:07,590 --> 00:25:10,180 And this the probability that you get that. 531 00:25:10,180 --> 00:25:13,060 So is this just a formula for saying what 532 00:25:13,060 --> 00:25:16,920 you're probability would be. 533 00:25:16,920 --> 00:25:21,450 So if I give you this, would that be a little easier? 534 00:25:21,450 --> 00:25:24,560 You can figure out what n is, what k is. 535 00:25:24,560 --> 00:25:27,705 AUDIENCE: Wait, k is 0.01-- no, p is 0.01, right? 536 00:25:27,705 --> 00:25:28,455 VINA NGUYEN: Yeah. 537 00:25:31,200 --> 00:25:34,318 So p is 0.01. 538 00:25:34,318 --> 00:25:37,360 AUDIENCE: I'm assuming n is 1,000 because n minus [? 10 ?] 539 00:25:37,360 --> 00:25:38,264 needs to be positive. 540 00:25:38,264 --> 00:25:38,716 VINA NGUYEN: Mm-hmm. 541 00:25:38,716 --> 00:25:40,120 AUDIENCE: So then [INAUDIBLE]. 542 00:25:40,120 --> 00:25:45,020 VINA NGUYEN: So k equals [? n0 ?] to-- 543 00:25:47,970 --> 00:25:48,740 AUDIENCE: 49? 544 00:25:48,740 --> 00:25:49,490 VINA NGUYEN: Yeah. 545 00:25:52,358 --> 00:25:52,983 AUDIENCE: Wait. 546 00:25:52,983 --> 00:25:55,478 Wouldn't it be 1 to 50? 547 00:25:59,470 --> 00:26:00,642 VINA NGUYEN: I think. 548 00:26:00,642 --> 00:26:02,900 I have to double check that. 549 00:26:02,900 --> 00:26:06,500 I think it's right. 550 00:26:06,500 --> 00:26:08,800 So-- OK, let me read this again. 551 00:26:08,800 --> 00:26:13,060 So you have 50 modems for 1,000 customers, 552 00:26:13,060 --> 00:26:18,140 but you can't have a k greater than 50, 553 00:26:18,140 --> 00:26:22,190 because you only have 50, right? 554 00:26:22,190 --> 00:26:22,829 AUDIENCE: Yeah. 555 00:26:22,829 --> 00:26:24,620 VINA NGUYEN: So that answers your question. 556 00:26:24,620 --> 00:26:27,540 AUDIENCE: So why can't you have 0? 557 00:26:27,540 --> 00:26:30,790 AUDIENCE: Because 0 isn't [INAUDIBLE].. 558 00:26:30,790 --> 00:26:34,150 Wait, which modem would be labeled number 0? 559 00:26:34,150 --> 00:26:36,550 It says there are 50 modems, so it would just be 1 to 50. 560 00:26:36,550 --> 00:26:39,020 AUDIENCE: Yeah, but you could be using none of the modems. 561 00:26:39,020 --> 00:26:40,353 VINA NGUYEN: Yeah, you're right. 562 00:26:40,353 --> 00:26:43,580 So you can-- like, maybe none of the customers need it. 563 00:26:43,580 --> 00:26:44,140 AUDIENCE: Oh. 564 00:26:44,140 --> 00:26:44,580 VINA NGUYEN: Yeah, yeah. 565 00:26:44,580 --> 00:26:45,271 You're right. 566 00:26:45,271 --> 00:26:45,770 Thank you. 567 00:26:45,770 --> 00:26:49,100 So it can go from 0 to 50. 568 00:26:49,100 --> 00:26:50,970 And then n is 1,000, because that's 569 00:26:50,970 --> 00:26:52,775 the total number of customers. 570 00:26:57,230 --> 00:26:59,210 OK, so that would be-- 571 00:27:12,075 --> 00:27:12,575 OK? 572 00:27:12,575 --> 00:27:14,560 So does that makes sense? 573 00:27:14,560 --> 00:27:17,957 If I give you this, is that easier? 574 00:27:17,957 --> 00:27:18,540 AUDIENCE: Yes. 575 00:27:22,000 --> 00:27:24,040 VINA NGUYEN: OK, so it's hard. 576 00:27:24,040 --> 00:27:24,940 OK. 577 00:27:24,940 --> 00:27:26,070 Do you guys understand? 578 00:27:26,070 --> 00:27:28,440 Any other questions?