1 00:00:05,060 --> 00:00:07,430 The following content is provided under a Creative 2 00:00:07,430 --> 00:00:08,820 Commons license. 3 00:00:08,820 --> 00:00:11,030 Your support will help MIT OpenCourseWare 4 00:00:11,030 --> 00:00:15,120 continue to offer high quality educational resources for free. 5 00:00:15,120 --> 00:00:17,660 To make a donation or to view additional materials 6 00:00:17,660 --> 00:00:21,620 from hundreds of MIT courses, visit MIT OpenCourseWare 7 00:00:21,620 --> 00:00:22,820 at ocw.mit.edu. 8 00:00:25,740 --> 00:00:29,489 VINA NGUYEN: So can anyone tell me what a random variable is, 9 00:00:29,489 --> 00:00:30,780 or do you want me to define it? 10 00:00:35,370 --> 00:00:40,940 AUDIENCE: Isn't it just any value within your sample space? 11 00:00:40,940 --> 00:00:42,220 VINA NGUYEN: Yeah. 12 00:00:42,220 --> 00:00:46,690 So it's a way to represent the value you want, 13 00:00:46,690 --> 00:00:47,680 which can be anything. 14 00:00:50,494 --> 00:00:51,940 I'll write it out. 15 00:01:35,510 --> 00:01:40,570 So remember how we talked about sample space? 16 00:01:40,570 --> 00:01:41,070 Oh, wait. 17 00:01:41,070 --> 00:01:41,935 What's your name? 18 00:01:41,935 --> 00:01:42,380 AUDIENCE: Eric. 19 00:01:42,380 --> 00:01:42,840 VINA NGUYEN: Eric. 20 00:01:42,840 --> 00:01:43,340 OK. 21 00:01:43,340 --> 00:01:45,030 Let me write you in. 22 00:01:45,030 --> 00:01:45,530 Eric. 23 00:01:51,940 --> 00:01:57,070 So imagine this is your sample space. 24 00:01:57,070 --> 00:02:00,310 So this is your universal sample space. 25 00:02:00,310 --> 00:02:04,480 And what a random variable does is take any of your outcomes 26 00:02:04,480 --> 00:02:08,740 in the sample space and puts it on a real number line. 27 00:02:08,740 --> 00:02:14,950 So this could be 0, 1, 2, 3. 28 00:02:14,950 --> 00:02:17,440 And say you have some random value here, 29 00:02:17,440 --> 00:02:21,380 you're going to map it to whatever real number value 30 00:02:21,380 --> 00:02:23,740 line that is. 31 00:02:23,740 --> 00:02:26,590 So the example that I have in here 32 00:02:26,590 --> 00:02:29,890 is if you say that your random variable x is 33 00:02:29,890 --> 00:02:40,310 the maximum of two rules, you have a sample space. 34 00:02:49,420 --> 00:02:51,230 So this is like your first roll. 35 00:02:51,230 --> 00:02:52,305 This is your second roll. 36 00:03:04,240 --> 00:03:05,820 So this is your sample space. 37 00:03:05,820 --> 00:03:08,470 And then you have the real number line, 38 00:03:08,470 --> 00:03:13,630 where x is some certain event. 39 00:03:13,630 --> 00:03:15,650 But x is the maximum to roll. 40 00:03:15,650 --> 00:03:23,950 So it could be like 1, 2, 3, 4, 5, 6. 41 00:03:23,950 --> 00:03:32,560 So if we roll the 2 and the 3, then that would be mapped to 3. 42 00:03:32,560 --> 00:03:34,790 Pretty straightforward. 43 00:03:34,790 --> 00:03:40,970 If you have 5 and 5, this sample space maps the 5, et cetera. 44 00:03:40,970 --> 00:03:42,220 Does everyone understand that? 45 00:03:42,220 --> 00:03:45,112 AUDIENCE: Oh, maximum roll size and pick one, 46 00:03:45,112 --> 00:03:46,558 and you've got to combine them? 47 00:03:46,558 --> 00:03:47,225 VINA NGUYEN: Hm? 48 00:03:47,225 --> 00:03:49,266 AUDIENCE: Oh, so when you meant max of two rolls, 49 00:03:49,266 --> 00:03:50,986 you meant the highest of the two rolls, 50 00:03:50,986 --> 00:03:51,930 not the sum of the rolls. 51 00:03:51,930 --> 00:03:52,190 VINA NGUYEN: Yeah. 52 00:03:52,190 --> 00:03:53,144 AUDIENCE: Oh. 53 00:03:53,144 --> 00:03:56,006 AUDIENCE: Is it also, like, [INAUDIBLE]?? 54 00:04:00,557 --> 00:04:01,390 VINA NGUYEN: Oh, OK. 55 00:04:01,390 --> 00:04:01,889 Sorry. 56 00:04:01,889 --> 00:04:06,300 So this is not really a coordinate system. 57 00:04:06,300 --> 00:04:09,940 It's just the way that we represent it. 58 00:04:12,700 --> 00:04:16,610 So I'm not mapping x against y. 59 00:04:16,610 --> 00:04:20,000 So x is separate from this sample space. 60 00:04:20,000 --> 00:04:24,410 So x can be any one of these numbers, 61 00:04:24,410 --> 00:04:28,900 but lower case x means a certain one of these numbers. 62 00:04:28,900 --> 00:04:33,720 So that's the difference between capitalized X and a lower 63 00:04:33,720 --> 00:04:37,540 case x, which is one of the questions you guys had. 64 00:04:37,540 --> 00:04:39,078 So does that make sense? 65 00:04:39,078 --> 00:04:42,010 OK. 66 00:04:42,010 --> 00:04:43,870 If you notice here, this is discrete 67 00:04:43,870 --> 00:04:50,030 because you can't have 1.1, 1.2, 1.3, 1.333, et cetera. 68 00:04:50,030 --> 00:04:51,470 So this is discrete. 69 00:04:51,470 --> 00:04:54,550 It's countable, and it's finite. 70 00:04:54,550 --> 00:04:58,630 So you can't have an infinite number in order to be discrete. 71 00:05:03,880 --> 00:05:07,400 If you look on the back of the first page, 72 00:05:07,400 --> 00:05:12,620 I already told you that the first three are discrete. 73 00:05:12,620 --> 00:05:15,380 So I'm just going to ask why the fourth and fifth 74 00:05:15,380 --> 00:05:17,980 random variables aren't discrete, 75 00:05:17,980 --> 00:05:19,310 if anyone can tell me that. 76 00:05:30,680 --> 00:05:32,470 Anybody? 77 00:05:32,470 --> 00:05:32,970 Yep? 78 00:05:32,970 --> 00:05:38,989 AUDIENCE: Well, the range is infinite for time. 79 00:05:38,989 --> 00:05:39,780 VINA NGUYEN: Right. 80 00:05:39,780 --> 00:05:40,740 Yeah. 81 00:05:40,740 --> 00:05:42,720 Did everyone hear that? 82 00:05:42,720 --> 00:05:44,880 So the range is infinite, and you can't discretely 83 00:05:44,880 --> 00:05:45,510 count them. 84 00:05:45,510 --> 00:05:46,110 Yep. 85 00:05:46,110 --> 00:05:47,550 Exactly. 86 00:05:47,550 --> 00:05:49,860 But the nice thing about random variables 87 00:05:49,860 --> 00:05:52,800 is that we can take a continuous thing like that 88 00:05:52,800 --> 00:05:54,390 and make it into discrete. 89 00:05:54,390 --> 00:05:58,550 So the example I have is, let's say 90 00:05:58,550 --> 00:06:00,210 a is some random variable that has 91 00:06:00,210 --> 00:06:04,880 a range from negative infinity to infinity. 92 00:06:04,880 --> 00:06:07,490 Does everyone understand this notation? 93 00:06:07,490 --> 00:06:08,490 OK. 94 00:06:08,490 --> 00:06:12,360 So we can't have a discrete random variable 95 00:06:12,360 --> 00:06:16,560 that describes this because it is continuous and infinite. 96 00:06:16,560 --> 00:06:18,990 So we're going to make a function. 97 00:06:18,990 --> 00:06:24,132 I just call it f because sgn confuse you guys. 98 00:06:24,132 --> 00:06:27,630 So we're going to convert a into discrete. 99 00:06:27,630 --> 00:06:33,150 So we're going to make it 1 if a is greater than 0, 0 100 00:06:33,150 --> 00:06:42,990 if a is equal to 0, and negative 1 if a is less than 0. 101 00:06:42,990 --> 00:06:47,970 So by taking a continuous sample space we've made it discrete. 102 00:06:47,970 --> 00:06:50,640 Does everyone see that? 103 00:06:50,640 --> 00:06:55,710 So you have this level of infinity. 104 00:06:55,710 --> 00:07:03,310 And we've mapped it into discrete values, 0, 1, 1. 105 00:07:03,310 --> 00:07:06,960 So does everyone see how we can use this random variable thing 106 00:07:06,960 --> 00:07:10,550 to make continuous things discrete? 107 00:07:10,550 --> 00:07:12,730 OK. 108 00:07:12,730 --> 00:07:18,660 So the thing about random variables 109 00:07:18,660 --> 00:07:21,400 is that they need probability mass functions, 110 00:07:21,400 --> 00:07:24,660 and that's just a very technical way of saying, 111 00:07:24,660 --> 00:07:29,010 what's the probability of x being any of these? 112 00:07:29,010 --> 00:07:34,995 So the way we write that is p of x. 113 00:07:50,800 --> 00:07:53,470 So what this is saying is the probability 114 00:07:53,470 --> 00:07:57,850 of this random variable x being a specific x. 115 00:07:57,850 --> 00:07:59,950 So the probability of x max of two 116 00:07:59,950 --> 00:08:04,290 rolls being specifically 1, 2, 3, 4, 5, 6. 117 00:08:04,290 --> 00:08:07,885 So that would be like this. 118 00:08:11,080 --> 00:08:13,400 That's basically what it's saying. 119 00:08:13,400 --> 00:08:18,660 So the example I have for this is 120 00:08:18,660 --> 00:08:24,330 if we say a random variable x-- this is just random variable. 121 00:08:24,330 --> 00:08:31,136 x equals the number of heads obtained. 122 00:08:46,280 --> 00:08:49,090 So if we have our random variable defined 123 00:08:49,090 --> 00:08:52,330 as x equals the number of heads obtained in a two 124 00:08:52,330 --> 00:08:55,360 toss sequence, the first thing we need to do 125 00:08:55,360 --> 00:08:57,430 is write down what our PMF is. 126 00:08:57,430 --> 00:09:00,030 What is our probability mass function? 127 00:09:00,030 --> 00:09:06,284 So what is this, essentially? 128 00:09:10,460 --> 00:09:13,500 Does anyone have any idea how to start? 129 00:09:13,500 --> 00:09:15,300 AUDIENCE: You could draw the table 130 00:09:15,300 --> 00:09:17,300 of all the possible outcomes. 131 00:09:17,300 --> 00:09:20,300 So you could have head-head, head-tail, tail-head, 132 00:09:20,300 --> 00:09:21,592 tail-tail. 133 00:09:21,592 --> 00:09:22,300 VINA NGUYEN: Mhm. 134 00:09:29,800 --> 00:09:30,680 Is that it? 135 00:09:30,680 --> 00:09:31,220 Oh, wait. 136 00:09:31,220 --> 00:09:31,470 You said-- 137 00:09:31,470 --> 00:09:32,636 AUDIENCE: Oh, and tail-tail. 138 00:09:32,636 --> 00:09:34,550 VINA NGUYEN: Yeah. 139 00:09:34,550 --> 00:09:42,710 So basically, x equals 0, x equals 1, x equals 2. 140 00:09:42,710 --> 00:09:44,240 Does everyone see that. 141 00:09:44,240 --> 00:09:47,780 So this means 0 heads, which would be this. 142 00:09:47,780 --> 00:09:52,160 1 should be these, and then 2, it should be that. 143 00:09:52,160 --> 00:09:58,862 So we're going to ask, what's the probability that x is 0? 144 00:09:58,862 --> 00:10:00,284 AUDIENCE: 1/4. 145 00:10:00,284 --> 00:10:01,117 VINA NGUYEN: Louder. 146 00:10:01,117 --> 00:10:02,020 AUDIENCE: 1/4. 147 00:10:02,020 --> 00:10:04,310 VINA NGUYEN: Yep. 148 00:10:04,310 --> 00:10:05,151 And 1. 149 00:10:05,151 --> 00:10:05,734 AUDIENCE: 1/2. 150 00:10:09,670 --> 00:10:13,120 AUDIENCE: [INAUDIBLE] 151 00:10:13,120 --> 00:10:15,200 VINA NGUYEN: And in probability mass functions 152 00:10:15,200 --> 00:10:16,380 we also want to add 0. 153 00:10:21,860 --> 00:10:23,295 So my TA got me for that a lot, so 154 00:10:23,295 --> 00:10:24,920 make sure you remember that in college. 155 00:10:28,190 --> 00:10:30,050 And the example here, though, I've 156 00:10:30,050 --> 00:10:32,210 combined these two, which is how you're technically 157 00:10:32,210 --> 00:10:33,570 supposed to do it. 158 00:10:33,570 --> 00:10:38,632 So we would actually take this out and then put that. 159 00:10:38,632 --> 00:10:40,340 So it's just a simpler way of writing it. 160 00:10:44,101 --> 00:10:44,600 See that? 161 00:10:51,720 --> 00:10:52,860 Am I going too fast? 162 00:10:52,860 --> 00:10:53,976 Does everyone understand? 163 00:11:00,280 --> 00:11:00,780 All right. 164 00:11:00,780 --> 00:11:04,190 So that's just an example of how to calculate PMFs. 165 00:11:04,190 --> 00:11:08,630 And if you notice, if you add these, you get 1. 166 00:11:08,630 --> 00:11:11,720 So it's easy to tell if you have this written out. 167 00:11:11,720 --> 00:11:14,700 1/4 plus 1/2 plus 1/4 is 1. 168 00:11:14,700 --> 00:11:17,240 So make sure, if you do write it in this way, 169 00:11:17,240 --> 00:11:20,090 that you add it twice, because you have two separate x's. 170 00:11:32,951 --> 00:11:35,810 So now we're going to talk about specific kinds 171 00:11:35,810 --> 00:11:37,350 of random variables. 172 00:11:37,350 --> 00:11:44,380 The first one is the easiest one, and that is the Bernoulli. 173 00:11:44,380 --> 00:11:44,880 Yeah. 174 00:11:44,880 --> 00:11:46,730 Bernoulli random variable. 175 00:11:59,180 --> 00:12:02,750 So essentially, this is your coin toss. 176 00:12:02,750 --> 00:12:05,980 So you have x equals heads. 177 00:12:09,440 --> 00:12:12,080 Or, in more general terms, that would be success. 178 00:12:17,540 --> 00:12:20,110 So this is your random variable, and that's 179 00:12:20,110 --> 00:12:22,040 how you're defining it. 180 00:12:22,040 --> 00:12:33,040 So your PMF is pretty simple. 181 00:12:33,040 --> 00:12:35,780 p equals the probability that you get heads, 182 00:12:35,780 --> 00:12:38,392 and then 1 minus p, probability that you don't. 183 00:12:38,392 --> 00:12:40,225 And we're going to use the real number line. 184 00:12:44,080 --> 00:12:46,370 So you have your sample space again. 185 00:12:46,370 --> 00:12:47,740 You have heads here. 186 00:12:47,740 --> 00:12:49,510 Tails. 187 00:12:49,510 --> 00:12:51,760 Even though it's not continuous, this doesn't actually 188 00:12:51,760 --> 00:12:53,470 have a real number line value, so we're 189 00:12:53,470 --> 00:12:58,450 going to make heads 1 and tails 0. 190 00:12:58,450 --> 00:13:03,440 So this is x equals 1 and 0. 191 00:13:07,520 --> 00:13:10,950 Does everyone see that's pretty simple? 192 00:13:10,950 --> 00:13:13,814 There's applications for this, like whether a telephone is 193 00:13:13,814 --> 00:13:15,730 free or busy for someone who is a telemarketer 194 00:13:15,730 --> 00:13:18,290 and wants to know the probability that the person 195 00:13:18,290 --> 00:13:22,160 will pick up, or if a person is sick or healthy, 196 00:13:22,160 --> 00:13:25,370 simple things like that. 197 00:13:25,370 --> 00:13:33,370 So our second one is the binomial random variable, 198 00:13:33,370 --> 00:13:37,460 which is basically a sequence of Bernoulli random variables. 199 00:13:49,140 --> 00:13:53,100 So the way this is set up is that you toss a coin N times, 200 00:13:53,100 --> 00:13:56,880 and then your random variable x is the number 201 00:13:56,880 --> 00:13:59,310 of times the heads comes up. 202 00:13:59,310 --> 00:14:07,930 So a number of heads in an N task sequence. 203 00:14:12,852 --> 00:14:14,310 The more general way of saying that 204 00:14:14,310 --> 00:14:17,055 is the number of successes in N number of trials. 205 00:14:30,030 --> 00:14:30,730 OK. 206 00:14:30,730 --> 00:14:33,970 And your P, again, is the probability 207 00:14:33,970 --> 00:14:35,540 of success in just one try. 208 00:14:44,430 --> 00:14:47,120 And that's where your Bernoulli comes in. 209 00:14:53,389 --> 00:14:54,430 So that's the definition. 210 00:14:54,430 --> 00:14:56,280 Does everyone understand how that's working? 211 00:14:56,280 --> 00:14:57,310 You flip once. 212 00:14:57,310 --> 00:14:57,850 What is it? 213 00:14:57,850 --> 00:14:59,810 You flip again, tails. 214 00:14:59,810 --> 00:15:01,060 Flip, heads. 215 00:15:01,060 --> 00:15:04,810 So in here, your x would be 2. 216 00:15:04,810 --> 00:15:07,910 That's one of the examples. 217 00:15:07,910 --> 00:15:12,420 So we're going to figure out what the PMF is. 218 00:15:12,420 --> 00:15:14,210 Have you guys seen this before? 219 00:15:14,210 --> 00:15:14,710 No? 220 00:15:14,710 --> 00:15:15,210 OK. 221 00:15:18,730 --> 00:15:21,340 We're going to use a lot of concepts 222 00:15:21,340 --> 00:15:25,630 to figure out what this is based on what you guys already know. 223 00:15:25,630 --> 00:15:27,776 You already know what coin tossing is. 224 00:15:39,734 --> 00:15:41,650 You guys know what multiplication rule is too, 225 00:15:41,650 --> 00:15:42,030 right? 226 00:15:42,030 --> 00:15:43,280 You have a sequence of things. 227 00:15:43,280 --> 00:15:47,470 You just multiply the probabilities. 228 00:15:47,470 --> 00:15:48,910 You know what combinations are. 229 00:15:53,350 --> 00:15:55,379 Does order matter in this or not? 230 00:15:55,379 --> 00:15:55,920 AUDIENCE: No. 231 00:15:55,920 --> 00:15:56,630 VINA NGUYEN: No. 232 00:15:56,630 --> 00:15:57,129 OK. 233 00:15:57,129 --> 00:15:58,214 Good. 234 00:15:58,214 --> 00:15:59,630 And then the fourth thing you just 235 00:15:59,630 --> 00:16:04,260 learned is random variables. 236 00:16:09,940 --> 00:16:11,810 Coin toss thing I will write up again. 237 00:16:11,810 --> 00:16:14,220 It's basically Bernoulli random variable. 238 00:16:14,220 --> 00:16:15,911 P, probability that you get heads, 239 00:16:15,911 --> 00:16:17,910 and 1 minus P is the probability that you don't. 240 00:16:21,310 --> 00:16:29,910 So for the multiplication rule, if we say we toss it 241 00:16:29,910 --> 00:16:40,420 N times and you get K number of heads. 242 00:16:46,480 --> 00:16:54,280 And for my example, we'll use N equals 5, K equals 2. 243 00:16:54,280 --> 00:16:59,479 So what's the probability that you get K heads? 244 00:16:59,479 --> 00:17:01,520 Actually, how would you calculate the probability 245 00:17:01,520 --> 00:17:04,470 that you get K heads N times? 246 00:17:04,470 --> 00:17:18,440 So you get-- so the probability of 1 is p. 247 00:17:18,440 --> 00:17:20,089 Just one trial. 248 00:17:20,089 --> 00:17:22,691 And then multiplication rule, you multiply this. 249 00:17:22,691 --> 00:17:23,690 That's what the star is. 250 00:17:27,365 --> 00:17:31,660 You do that again, but this time it's 1 minus p. 251 00:17:31,660 --> 00:17:34,480 1 minus p. 252 00:17:34,480 --> 00:17:35,590 1 minus p. 253 00:17:38,270 --> 00:17:41,210 And the only reason you can do this is because of this. 254 00:17:41,210 --> 00:17:42,950 So you know that. 255 00:17:42,950 --> 00:17:49,810 And another way to write this is p K times, 256 00:17:49,810 --> 00:17:52,430 because K is the number of times you got heads. 257 00:17:52,430 --> 00:17:59,300 And then 1 minus p, N minus K. Does everyone see that? 258 00:17:59,300 --> 00:18:01,950 Because N is the total, and K is the number of heads. 259 00:18:01,950 --> 00:18:05,790 So you just want to take the difference. 260 00:18:05,790 --> 00:18:09,070 So that's part 2. 261 00:18:09,070 --> 00:18:10,990 For combinations, you know that's not 262 00:18:10,990 --> 00:18:15,640 the only way you can get two heads and three tails. 263 00:18:15,640 --> 00:18:27,490 You can have that, you can have this, et cetera, et cetera. 264 00:18:27,490 --> 00:18:29,360 So your combinations comes in. 265 00:18:29,360 --> 00:18:32,350 And how many of these sequences are possible? 266 00:18:32,350 --> 00:18:35,130 And we learned that's this many. 267 00:18:37,720 --> 00:18:39,410 So that would be 5, 2. 268 00:18:48,840 --> 00:18:50,440 Does everyone see that? 269 00:18:50,440 --> 00:18:53,770 And of course, you times it by this probability. 270 00:18:53,770 --> 00:18:57,250 So this is the number of times you can have this combination, 271 00:18:57,250 --> 00:19:00,910 where K is the number of heads and N is the number of tosses. 272 00:19:05,955 --> 00:19:06,830 Does that make sense? 273 00:19:10,420 --> 00:19:13,970 So you combine them all together, 274 00:19:13,970 --> 00:19:18,560 and you get probability of x equals 275 00:19:18,560 --> 00:19:21,040 K. We're going to use K in this example 276 00:19:21,040 --> 00:19:28,260 so that differentiates it with N. 277 00:19:28,260 --> 00:19:32,050 Equals the number of combinations times 278 00:19:32,050 --> 00:19:35,020 the probability, which we got there. 279 00:19:41,962 --> 00:19:42,920 Does everyone see that? 280 00:19:42,920 --> 00:19:43,419 Sorry. 281 00:19:47,570 --> 00:19:49,830 So what's our restriction on K? 282 00:19:53,060 --> 00:19:56,210 It can't be this, right? 283 00:19:56,210 --> 00:19:57,670 That doesn't work. 284 00:19:57,670 --> 00:19:59,961 So what does K have to be? 285 00:19:59,961 --> 00:20:02,810 AUDIENCE: [INAUDIBLE] 286 00:20:02,810 --> 00:20:03,560 VINA NGUYEN: Yeah. 287 00:20:03,560 --> 00:20:04,975 And also 0. 288 00:20:04,975 --> 00:20:06,440 It has an N of 0. 289 00:20:10,150 --> 00:20:12,990 And this is discrete again. 290 00:20:12,990 --> 00:20:13,931 Can you see? 291 00:20:13,931 --> 00:20:14,430 OK. 292 00:20:27,010 --> 00:20:29,290 So those are two of the major random variables 293 00:20:29,290 --> 00:20:31,990 that people usually start teaching with. 294 00:20:31,990 --> 00:20:33,640 So does everyone understand that? 295 00:20:36,320 --> 00:20:38,696 I'm going to erase this. 296 00:20:38,696 --> 00:20:39,690 Where's the eraser? 297 00:20:45,214 --> 00:20:46,130 Do you guys need this? 298 00:21:15,220 --> 00:21:19,100 So if you just have equations like this 299 00:21:19,100 --> 00:21:21,350 it's kind of hard to get a feel for what 300 00:21:21,350 --> 00:21:23,330 your random variable is saying. 301 00:21:23,330 --> 00:21:26,480 So we're going to graph it. 302 00:21:26,480 --> 00:21:28,660 And this is called the distribution. 303 00:21:36,330 --> 00:21:39,240 This is your coin toss. 304 00:21:43,260 --> 00:21:45,860 This is the x's that it can take, 305 00:21:45,860 --> 00:21:55,840 so 0, 1, and this is your PMF which we calculated. 306 00:21:55,840 --> 00:22:00,800 So the probability that you get 0 is 1 minus p let's say. 307 00:22:06,920 --> 00:22:10,320 And this could be p or something. 308 00:22:10,320 --> 00:22:13,930 And this can be reversed depending on what p is. 309 00:22:13,930 --> 00:22:17,760 So for your binomial-- 310 00:22:36,240 --> 00:22:43,950 so for your binomial, if I say that N equals 9 and p equals 311 00:22:43,950 --> 00:22:47,260 1/2, then it's going to be symmetrical. 312 00:22:47,260 --> 00:22:54,750 So if you have 9, 0, it's going to look like whatever number 313 00:22:54,750 --> 00:22:55,500 that is. 314 00:22:55,500 --> 00:22:56,230 4 and 5. 315 00:23:04,040 --> 00:23:06,600 So symmetrical of p equals 1/2. 316 00:23:06,600 --> 00:23:11,160 But if p is less than 1/2, it's going to be skewed this way. 317 00:23:11,160 --> 00:23:12,924 And if p is greater than 1/2, it's 318 00:23:12,924 --> 00:23:14,090 going to be skewed that way. 319 00:23:34,764 --> 00:23:36,930 Does everyone see that it's really easy to calculate 320 00:23:36,930 --> 00:23:37,930 if you do have numbers. 321 00:23:37,930 --> 00:23:41,620 You just plug in x, plot the probability, 322 00:23:41,620 --> 00:23:44,600 and then you get this distribution. 323 00:23:44,600 --> 00:23:48,550 So these are all called distributions. 324 00:23:55,585 --> 00:23:58,530 Does that makes sense? 325 00:23:58,530 --> 00:24:05,272 So so far we know the Bernoulli random variable 326 00:24:05,272 --> 00:24:05,980 and the binomial. 327 00:24:11,110 --> 00:24:15,980 So I'm going to tell you about the geometric random variable. 328 00:24:19,610 --> 00:24:21,740 And the way the random variable is 329 00:24:21,740 --> 00:24:25,610 described for that is x equals the number of tosses 330 00:24:25,610 --> 00:24:28,655 needed for a head to come up. 331 00:24:28,655 --> 00:24:29,780 AUDIENCE: Do you need this? 332 00:24:29,780 --> 00:24:33,050 VINA NGUYEN: No, you can-- you can just put it there. 333 00:24:33,050 --> 00:24:38,060 So that x includes the toss where you get a head. 334 00:24:38,060 --> 00:24:57,210 So number of tosses needed for a head to come up the first time. 335 00:25:04,500 --> 00:25:06,930 AUDIENCE: Why the first time? 336 00:25:06,930 --> 00:25:09,180 VINA NGUYEN: It's just like if you were simulating 337 00:25:09,180 --> 00:25:11,263 how many times you need to do something before you 338 00:25:11,263 --> 00:25:13,950 get it the first shot. 339 00:25:13,950 --> 00:25:14,790 AUDIENCE: Oh, OK. 340 00:25:14,790 --> 00:25:15,540 VINA NGUYEN: Yeah. 341 00:25:29,690 --> 00:25:41,050 So if we say that K is the number of times, 342 00:25:41,050 --> 00:25:44,334 then how would we write that probability? 343 00:25:57,180 --> 00:26:01,937 So this would be K equals 5. 344 00:26:01,937 --> 00:26:03,020 So it's kind of like that. 345 00:26:03,020 --> 00:26:05,544 You have p, p1 minus p, et cetera. 346 00:26:08,150 --> 00:26:11,815 Probability that you don't get heads for how many times? 347 00:26:11,815 --> 00:26:12,599 AUDIENCE: One. 348 00:26:12,599 --> 00:26:13,390 VINA NGUYEN: Right. 349 00:26:13,390 --> 00:26:17,190 Which is K minus 1. 350 00:26:17,190 --> 00:26:19,180 And the probability that you get it once? 351 00:26:19,180 --> 00:26:22,440 It is just 1. 352 00:26:22,440 --> 00:26:23,410 Does everyone see that? 353 00:26:23,410 --> 00:26:25,830 So that is how you write your PMF. 354 00:26:25,830 --> 00:26:35,270 So the PMF for this random variable 355 00:26:35,270 --> 00:26:44,170 equals 1 minus p, K minus 1 p. 356 00:26:44,170 --> 00:26:47,318 And in this case, what can K equal? 357 00:26:47,318 --> 00:26:49,570 It can't be 0 this time. 358 00:26:49,570 --> 00:26:50,908 So it starts from 1. 359 00:26:50,908 --> 00:26:53,697 AUDIENCE: Goes to K, right? 360 00:26:53,697 --> 00:26:55,780 VINA NGUYEN: No, because you're defining K, right? 361 00:26:59,240 --> 00:27:00,950 OK. 362 00:27:00,950 --> 00:27:04,730 So it's countable, which is OK, which makes it still discrete, 363 00:27:04,730 --> 00:27:07,910 even though it does go to infinity. 364 00:27:07,910 --> 00:27:13,350 So that would be discrete. 365 00:27:17,750 --> 00:27:21,500 And if we graph it to give you more visual understanding 366 00:27:21,500 --> 00:27:33,540 of what this looks like, if it only takes one time, what's 367 00:27:33,540 --> 00:27:35,265 the probability? 368 00:27:35,265 --> 00:27:35,765 p. 369 00:27:35,765 --> 00:27:36,265 Right. 370 00:27:41,290 --> 00:27:43,630 And if you graph it, it will slowly 371 00:27:43,630 --> 00:27:51,145 go down, like this, depending on how you choose p and what K is, 372 00:27:51,145 --> 00:27:52,401 et cetera. 373 00:27:52,401 --> 00:27:54,400 So that's just an example of what it looks like. 374 00:27:56,990 --> 00:27:59,899 So applications for this could be the number 375 00:27:59,899 --> 00:28:01,690 of times you need to take a test before you 376 00:28:01,690 --> 00:28:08,530 pass if your probability is, like, 0.6, which is not good. 377 00:28:08,530 --> 00:28:11,980 Another example could be finding a missing item in a given 378 00:28:11,980 --> 00:28:12,574 search. 379 00:28:12,574 --> 00:28:13,990 So that could be like an airplane, 380 00:28:13,990 --> 00:28:15,739 where they're trying to find your luggage, 381 00:28:15,739 --> 00:28:18,790 where p is like 0.0001. 382 00:28:18,790 --> 00:28:21,370 But that's some real life examples, 383 00:28:21,370 --> 00:28:23,120 because no one really cares about heads. 384 00:28:23,120 --> 00:28:23,620 OK. 385 00:28:23,620 --> 00:28:26,580 Does anyone need this? 386 00:28:26,580 --> 00:28:27,361 Anyone need this? 387 00:28:27,361 --> 00:28:27,860 OK. 388 00:28:35,460 --> 00:28:38,610 The fourth random variable is a little bit more complicated. 389 00:28:38,610 --> 00:28:40,600 Does everyone know what e is? 390 00:28:40,600 --> 00:28:42,290 Natural number. 391 00:28:42,290 --> 00:28:43,100 Two point whatever. 392 00:28:48,990 --> 00:28:54,634 So now you know a third kind, geometric. 393 00:28:54,634 --> 00:28:56,300 And the fourth one I'm going to tell you 394 00:28:56,300 --> 00:28:59,660 about today is the Poisson kind of variable. 395 00:29:08,480 --> 00:29:11,213 So has anyone heard of this before? 396 00:29:11,213 --> 00:29:12,594 AUDIENCE: I've heard of it. 397 00:29:12,594 --> 00:29:14,010 VINA NGUYEN: So I'm going tell you 398 00:29:14,010 --> 00:29:16,940 the PMF right off the bat instead of deriving it. 399 00:29:54,130 --> 00:29:56,380 So the Poisson random variable is mainly 400 00:29:56,380 --> 00:29:59,410 used to approximate binomials. 401 00:29:59,410 --> 00:30:03,430 And you know what binomial RVs are anyway. 402 00:30:03,430 --> 00:30:08,470 So K is the number of heads in an N toss sequence. 403 00:30:08,470 --> 00:30:11,890 So this works only if this lambda here 404 00:30:11,890 --> 00:30:16,120 is equal to Np, where N is the number of tosses 405 00:30:16,120 --> 00:30:18,730 and p is the probability of success. 406 00:30:18,730 --> 00:30:23,770 And N has to be really large, and p has to be very small. 407 00:30:26,810 --> 00:30:30,430 So instead of a coin toss, where you could have N equals 10 408 00:30:30,430 --> 00:30:32,950 and p equals 1/2, p has to be like-- 409 00:30:32,950 --> 00:30:35,800 say it's 0.01, which is really small, 410 00:30:35,800 --> 00:30:39,706 and this could be like 1,000. 411 00:30:39,706 --> 00:30:41,420 They're relative to each other. 412 00:30:47,750 --> 00:30:52,630 So an example of this could be if you're 413 00:30:52,630 --> 00:30:54,460 fixing the number of typos in a book 414 00:30:54,460 --> 00:30:56,470 and your probability is really small, 415 00:30:56,470 --> 00:30:58,960 but your N is large because N could be the number of words, 416 00:30:58,960 --> 00:31:01,520 which is a lot. 417 00:31:01,520 --> 00:31:04,000 Or another example would be the number 418 00:31:04,000 --> 00:31:07,000 of cars that get into accidents everyday, where N is like-- 419 00:31:07,000 --> 00:31:08,040 AUDIENCE: [INAUDIBLE]. 420 00:31:08,040 --> 00:31:08,789 VINA NGUYEN: Yeah. 421 00:31:08,789 --> 00:31:13,561 Where N is a lot and p is, hopefully, pretty small. 422 00:31:13,561 --> 00:31:15,550 AUDIENCE: [INAUDIBLE] 423 00:31:15,550 --> 00:31:17,350 VINA NGUYEN: So we're going to do a problem 424 00:31:17,350 --> 00:31:19,972 to help you understand. 425 00:31:19,972 --> 00:31:20,680 Anyone need this? 426 00:31:40,024 --> 00:31:42,510 AUDIENCE: What page are we? 427 00:31:42,510 --> 00:31:44,280 VINA NGUYEN: You're page-- 428 00:31:44,280 --> 00:31:45,090 second. 429 00:31:45,090 --> 00:31:46,861 Second. 430 00:31:46,861 --> 00:31:48,110 Does everyone see the problem? 431 00:31:48,110 --> 00:31:50,880 Because I really don't feel like writing it. 432 00:32:09,221 --> 00:32:11,964 Did everyone read it yet? 433 00:32:11,964 --> 00:32:13,340 AUDIENCE: Almost. 434 00:32:13,340 --> 00:32:16,170 VINA NGUYEN: Almost. 435 00:32:16,170 --> 00:32:19,860 It's called problem, and it's on the second page. 436 00:32:19,860 --> 00:32:21,960 [LAUGHTER] 437 00:32:22,300 --> 00:32:22,800 OK. 438 00:32:22,800 --> 00:32:25,000 Just checking. 439 00:32:25,000 --> 00:32:25,786 It's early for me. 440 00:32:25,786 --> 00:32:27,660 AUDIENCE: Exactly one birthday, how specific? 441 00:32:27,660 --> 00:32:28,680 Just, like, the day? 442 00:32:28,680 --> 00:32:32,492 So it has to be year. 443 00:32:32,492 --> 00:32:33,200 VINA NGUYEN: Yep. 444 00:32:36,020 --> 00:32:41,315 AUDIENCE: [INAUDIBLE] 445 00:32:41,315 --> 00:32:45,075 VINA NGUYEN: So there's only 365 days. 446 00:32:45,075 --> 00:32:46,440 Not 366. 447 00:32:46,440 --> 00:32:47,824 Just 365. 448 00:32:47,824 --> 00:32:52,930 AUDIENCE: 365. 449 00:32:52,930 --> 00:32:54,350 VINA NGUYEN: Round down. 450 00:32:54,350 --> 00:32:54,850 OK. 451 00:32:58,490 --> 00:33:00,510 Is everyone good? 452 00:33:00,510 --> 00:33:01,010 OK. 453 00:33:01,010 --> 00:33:01,885 So I'll summarize it. 454 00:33:01,885 --> 00:33:03,860 You have a party with 500 guests. 455 00:33:03,860 --> 00:33:06,380 What is the probability that only one guest 456 00:33:06,380 --> 00:33:08,060 has the same birthday as you? 457 00:33:08,060 --> 00:33:15,870 So we're going to solve it using the binomial way and then 458 00:33:15,870 --> 00:33:18,460 the Poisson approximation. 459 00:33:21,960 --> 00:33:27,350 So what's your N? 460 00:33:27,350 --> 00:33:30,714 What's the total number that we're going to add here? 461 00:33:30,714 --> 00:33:31,619 AUDIENCE: 500. 462 00:33:31,619 --> 00:33:32,577 VINA NGUYEN: Oh, sorry. 463 00:33:32,577 --> 00:33:34,200 500 includes yourself. 464 00:33:34,200 --> 00:33:34,700 So-- 465 00:33:34,700 --> 00:33:35,283 AUDIENCE: 499. 466 00:33:35,283 --> 00:33:36,530 VINA NGUYEN: Yeah. 467 00:33:36,530 --> 00:33:39,650 So 499. 468 00:33:39,650 --> 00:33:42,025 What's K? 469 00:33:42,025 --> 00:33:44,386 K is your success rate. 470 00:33:44,386 --> 00:33:46,060 AUDIENCE: 1. 471 00:33:46,060 --> 00:33:46,770 VINA NGUYEN: Yep. 472 00:33:46,770 --> 00:33:49,295 And p, the probability? 473 00:33:49,295 --> 00:33:51,291 AUDIENCE: 499. 474 00:33:51,291 --> 00:33:53,159 VINA NGUYEN: I can't hear. 475 00:33:53,159 --> 00:33:55,820 I can't hear. 476 00:33:55,820 --> 00:33:57,500 I can't hear you guys. 477 00:33:57,500 --> 00:33:58,520 1 out of-- 478 00:33:58,520 --> 00:33:59,337 AUDIENCE: 499. 479 00:33:59,337 --> 00:33:59,836 AUDIENCE: 3. 480 00:33:59,836 --> 00:34:00,968 VINA NGUYEN: No, 3. 481 00:34:00,968 --> 00:34:03,310 AUDIENCE: [INAUDIBLE] 482 00:34:03,310 --> 00:34:04,370 VINA NGUYEN: Yeah. 483 00:34:04,370 --> 00:34:05,870 AUDIENCE: Because you only have one. 484 00:34:05,870 --> 00:34:08,073 VINA NGUYEN: What were you saying before, Priya? 485 00:34:08,073 --> 00:34:09,500 Do you understand why? 486 00:34:09,500 --> 00:34:10,159 OK. 487 00:34:10,159 --> 00:34:12,200 So this is number of days, and you only want one. 488 00:34:17,249 --> 00:34:19,199 So here's your problem set up. 489 00:34:19,199 --> 00:34:21,750 If we do it the binomial way, how do we write that? 490 00:34:26,449 --> 00:34:30,679 You have N, K, et cetera. 491 00:34:30,679 --> 00:34:32,330 AUDIENCE: p, K. 492 00:34:32,330 --> 00:34:33,935 VINA NGUYEN: We'll just plug it in. 493 00:34:33,935 --> 00:34:35,800 AUDIENCE: Oh. 494 00:34:35,800 --> 00:34:40,399 499. 495 00:34:40,399 --> 00:34:44,809 And then 365. 496 00:34:44,809 --> 00:34:45,789 VINA NGUYEN: Mhm. 497 00:34:45,789 --> 00:34:58,070 AUDIENCE: [INAUDIBLE] 498 00:34:58,070 --> 00:34:59,430 VINA NGUYEN: 48. 499 00:34:59,430 --> 00:35:00,730 Right. 500 00:35:00,730 --> 00:35:09,230 N minus K. So what we're doing is just filling in this part. 501 00:35:09,230 --> 00:35:20,440 NK, PK, 1 minus p, N minus K. So this 502 00:35:20,440 --> 00:35:23,290 is kind of like binomial and geometric 503 00:35:23,290 --> 00:35:26,405 since we are just doing K equals 1. 504 00:35:34,480 --> 00:35:35,620 So how do we solve that? 505 00:35:35,620 --> 00:35:37,160 What does this become? 506 00:35:37,160 --> 00:35:38,140 AUDIENCE: 499. 507 00:35:38,140 --> 00:35:40,562 [INTERPOSING VOICES] 508 00:35:40,562 --> 00:35:41,312 VINA NGUYEN: What? 509 00:35:41,312 --> 00:35:42,854 AUDIENCE: It's sort of a little hard. 510 00:35:42,854 --> 00:35:43,603 VINA NGUYEN: Yeah. 511 00:35:43,603 --> 00:35:44,590 It's really hard. 512 00:35:44,590 --> 00:35:49,198 So you have 498. 513 00:35:49,198 --> 00:35:49,823 AUDIENCE: Yeah. 514 00:35:49,823 --> 00:35:52,090 So it's just 499. 515 00:35:52,090 --> 00:35:52,840 VINA NGUYEN: Yeah. 516 00:35:52,840 --> 00:35:54,490 But if you had a fair number. 517 00:35:54,490 --> 00:35:57,070 Yeah. 518 00:35:57,070 --> 00:35:57,900 This part is hard. 519 00:36:02,900 --> 00:36:06,400 AUDIENCE: [INAUDIBLE] 520 00:36:06,400 --> 00:36:08,779 VINA NGUYEN: Anyone tell me what this is? 521 00:36:08,779 --> 00:36:09,820 You're not understanding? 522 00:36:09,820 --> 00:36:10,630 OK. 523 00:36:10,630 --> 00:36:15,750 This is 0.3486. 524 00:36:15,750 --> 00:36:20,715 So this is what you get when you do it the exact way. 525 00:36:20,715 --> 00:36:24,950 AUDIENCE: [INAUDIBLE] 526 00:36:24,950 --> 00:36:25,701 VINA NGUYEN: Wait. 527 00:36:25,701 --> 00:36:26,200 Sorry. 528 00:36:26,200 --> 00:36:26,750 Sorry. 529 00:36:26,750 --> 00:36:27,140 Yeah. 530 00:36:27,140 --> 00:36:27,639 Wait, is it? 531 00:36:27,639 --> 00:36:28,372 AUDIENCE: Yep. 532 00:36:28,372 --> 00:36:29,510 Oh, yeah, that's right. 533 00:36:29,510 --> 00:36:31,115 VINA NGUYEN: That's right, isn't it? 534 00:36:31,115 --> 00:36:32,097 To calculate it. 535 00:36:32,097 --> 00:36:32,930 AUDIENCE: It's high. 536 00:36:32,930 --> 00:36:33,350 VINA NGUYEN: Yeah. 537 00:36:33,350 --> 00:36:34,016 Kind of figured. 538 00:36:34,016 --> 00:36:36,680 AUDIENCE: Well, we have so many people. 539 00:36:36,680 --> 00:36:44,810 VINA NGUYEN: So if we do it the Poisson way, what is this? 540 00:36:44,810 --> 00:36:46,972 This is our parameter. 541 00:36:46,972 --> 00:36:50,469 AUDIENCE: [INAUDIBLE] 542 00:36:50,469 --> 00:36:51,260 VINA NGUYEN: N is-- 543 00:36:57,428 --> 00:37:01,332 AUDIENCE: [INAUDIBLE] 544 00:37:01,332 --> 00:37:03,790 VINA NGUYEN: Before I go on, does everyone understand this? 545 00:37:03,790 --> 00:37:05,140 AUDIENCE: Yeah. 546 00:37:05,140 --> 00:37:06,420 VINA NGUYEN: OK. 547 00:37:06,420 --> 00:37:07,420 So you have-- 548 00:37:25,746 --> 00:37:26,870 I wrote this kind of funny. 549 00:37:48,470 --> 00:37:48,970 All right. 550 00:37:48,970 --> 00:37:51,010 So this is our PMF. 551 00:37:51,010 --> 00:37:53,350 So if you plug everything in, what do you get? 552 00:37:58,817 --> 00:38:04,284 AUDIENCE: [INAUDIBLE] 553 00:38:04,284 --> 00:38:07,669 AUDIENCE: [INAUDIBLE] 554 00:38:07,669 --> 00:38:08,335 VINA NGUYEN: Hm? 555 00:38:08,335 --> 00:38:09,657 AUDIENCE: What's K again? 556 00:38:09,657 --> 00:38:10,490 VINA NGUYEN: K is 1. 557 00:38:17,130 --> 00:38:17,740 Is that right? 558 00:38:17,740 --> 00:38:19,960 AUDIENCE: Yeah. 559 00:38:19,960 --> 00:38:22,660 VINA NGUYEN: So this has a lot less exponents and stuff, 560 00:38:22,660 --> 00:38:24,970 so it's a lot easier to calculate. 561 00:38:24,970 --> 00:38:25,780 So what do we get? 562 00:38:28,756 --> 00:38:34,708 AUDIENCE: [INAUDIBLE] 563 00:38:34,708 --> 00:38:35,700 VINA NGUYEN: This guy. 564 00:38:39,668 --> 00:38:44,660 AUDIENCE: Oh, 0.348. 565 00:38:44,660 --> 00:38:45,540 VINA NGUYEN: Yep. 566 00:38:45,540 --> 00:38:47,940 So you get a pretty good approximation. 567 00:38:52,090 --> 00:38:58,210 So by using the Poisson, you can get pretty good approximation 568 00:38:58,210 --> 00:39:01,160 without doing all this extra calculation. 569 00:39:01,160 --> 00:39:03,907 So that's one of the reasons they made this random variable. 570 00:39:03,907 --> 00:39:04,990 Well, they didn't make it. 571 00:39:04,990 --> 00:39:05,694 They derived it. 572 00:39:05,694 --> 00:39:07,610 AUDIENCE: So it's like an estimation variable. 573 00:39:07,610 --> 00:39:08,770 VINA NGUYEN: Yeah. 574 00:39:08,770 --> 00:39:11,931 For this specific application. 575 00:39:14,700 --> 00:39:17,640 Does that make sense to everybody? 576 00:39:17,640 --> 00:39:20,220 Keep in mind, this only works, again, if N is large 577 00:39:20,220 --> 00:39:21,180 and p is small. 578 00:39:21,180 --> 00:39:25,000 So you can't do it if it's just a coin toss. 579 00:39:25,000 --> 00:39:25,500 OK. 580 00:39:25,500 --> 00:39:28,070 AUDIENCE: What happens [INAUDIBLE] 581 00:39:28,070 --> 00:39:31,030 VINA NGUYEN: Just calculator doesn't work. 582 00:39:31,030 --> 00:39:32,610 So what's this? 583 00:39:32,610 --> 00:39:34,950 What's the Bernoulli one? 584 00:39:34,950 --> 00:39:37,350 AUDIENCE: Bernoulli is-- 585 00:39:37,350 --> 00:39:39,600 VINA NGUYEN: What is x? 586 00:39:39,600 --> 00:39:40,530 Our random variable. 587 00:39:40,530 --> 00:39:41,250 What is it? 588 00:39:41,250 --> 00:39:43,444 What are we defining it as? 589 00:39:43,444 --> 00:39:44,770 AUDIENCE: Success. 590 00:39:44,770 --> 00:39:48,210 VINA NGUYEN: In just one trial. 591 00:39:48,210 --> 00:39:48,960 AUDIENCE: Several. 592 00:39:48,960 --> 00:39:50,043 VINA NGUYEN: No, just one. 593 00:39:50,043 --> 00:39:50,980 AUDIENCE: Oh, right. 594 00:39:50,980 --> 00:39:52,470 VINA NGUYEN: Just one. 595 00:39:52,470 --> 00:39:55,631 This is N number of trials. 596 00:39:55,631 --> 00:39:57,464 What's this? 597 00:39:57,464 --> 00:39:58,925 AUDIENCE: How many times it takes. 598 00:39:58,925 --> 00:40:00,386 AUDIENCE: How long it takes. 599 00:40:00,386 --> 00:40:02,607 How many trials it takes in order to get success. 600 00:40:02,607 --> 00:40:04,440 VINA NGUYEN: And this is including the trial 601 00:40:04,440 --> 00:40:06,750 that you do get success. 602 00:40:06,750 --> 00:40:09,998 And Poisson is what you just learned. 603 00:40:09,998 --> 00:40:12,270 AUDIENCE: [INAUDIBLE] 604 00:40:12,270 --> 00:40:13,340 VINA NGUYEN: For that. 605 00:40:13,340 --> 00:40:14,300 OK. 606 00:40:14,300 --> 00:40:15,420 Good. 607 00:40:15,420 --> 00:40:18,780 So those are the four main random variables. 608 00:40:22,270 --> 00:40:28,600 And I'm going to tell you what expectation and variance is. 609 00:40:28,600 --> 00:40:30,400 Do you guys need any of this? 610 00:40:43,670 --> 00:40:46,580 So even though it's a funny name like random variable, 611 00:40:46,580 --> 00:40:49,850 it's basically just summarizing what you guys already know-- 612 00:40:49,850 --> 00:40:52,640 probabilities, sample space, et cetera. 613 00:40:52,640 --> 00:40:54,950 It's just a very short way of writing all that. 614 00:41:04,393 --> 00:41:05,884 Anyone need this? 615 00:41:20,794 --> 00:41:23,034 This? 616 00:41:23,034 --> 00:41:23,980 Still need it? 617 00:41:23,980 --> 00:41:24,480 OK. 618 00:41:34,610 --> 00:41:38,080 How many of you guys have taken statistics 619 00:41:38,080 --> 00:41:39,680 or any kind of statistics? 620 00:41:39,680 --> 00:41:41,370 AUDIENCE: [INAUDIBLE] 621 00:41:41,370 --> 00:41:43,320 VINA NGUYEN: Kind of? 622 00:41:43,320 --> 00:41:45,580 Do you guys know what mean is? 623 00:41:45,580 --> 00:41:46,080 OK. 624 00:41:46,080 --> 00:41:47,770 Do you know what variance is? 625 00:41:47,770 --> 00:41:48,444 AUDIENCE: No. 626 00:41:48,444 --> 00:41:49,110 VINA NGUYEN: No? 627 00:41:49,110 --> 00:41:50,030 OK. 628 00:41:50,030 --> 00:41:51,810 So I'll skim over mean then. 629 00:42:02,065 --> 00:42:04,100 So in probability, mean is actually 630 00:42:04,100 --> 00:42:07,300 called expectation, which is your expected value. 631 00:42:12,460 --> 00:42:17,770 And the reason for that is because a mean kind of implies 632 00:42:17,770 --> 00:42:19,760 a bunch of experiments and you find the mean. 633 00:42:19,760 --> 00:42:22,790 But in probability, you might only do one trial. 634 00:42:22,790 --> 00:42:24,730 So this is your expected value, even 635 00:42:24,730 --> 00:42:27,070 if it is the same thing, mathematically, 636 00:42:27,070 --> 00:42:29,820 as your average. 637 00:42:29,820 --> 00:42:34,300 And then variance is just another way 638 00:42:34,300 --> 00:42:39,065 of describing how spread out your data is from that mean. 639 00:42:39,065 --> 00:42:40,740 AUDIENCE: [INAUDIBLE] 640 00:42:40,740 --> 00:42:41,490 VINA NGUYEN: Yeah. 641 00:42:41,490 --> 00:42:44,400 AUDIENCE: Nice. 642 00:42:44,400 --> 00:42:47,795 [SIDE CONVERSATION] 643 00:42:53,130 --> 00:42:55,070 VINA NGUYEN: And like you've mentioned, 644 00:42:55,070 --> 00:42:58,100 you probably already know what standard deviation is. 645 00:42:58,100 --> 00:43:03,084 That's equal to the square root of variance. 646 00:43:03,084 --> 00:43:04,794 AUDIENCE: Oh. 647 00:43:04,794 --> 00:43:05,544 VINA NGUYEN: Yeah. 648 00:43:08,764 --> 00:43:10,430 So this is kind of easy because you guys 649 00:43:10,430 --> 00:43:11,804 sound like you already know this. 650 00:43:14,294 --> 00:43:15,354 AUDIENCE: Not really. 651 00:43:15,354 --> 00:43:16,020 VINA NGUYEN: No? 652 00:43:16,020 --> 00:43:17,502 I will keep on going. 653 00:43:23,133 --> 00:43:23,924 AUDIENCE: Standard. 654 00:43:23,924 --> 00:43:27,345 Is that STD? 655 00:43:27,345 --> 00:43:28,950 VINA NGUYEN: They're not all caps. 656 00:43:28,950 --> 00:43:31,405 [LAUGHTER] 657 00:43:34,514 --> 00:43:36,810 Is that better? 658 00:43:36,810 --> 00:43:37,770 All right. 659 00:43:37,770 --> 00:43:41,256 [SIDE CONVERSATION] 660 00:44:10,890 --> 00:44:13,290 So I'll run through this example to show you 661 00:44:13,290 --> 00:44:16,350 what expectation is. 662 00:44:16,350 --> 00:44:18,840 So you have two independent coin tosses, 663 00:44:18,840 --> 00:44:22,350 and the probability that you can heads is 3/4, 664 00:44:22,350 --> 00:44:24,330 and your random variable is x equals 665 00:44:24,330 --> 00:44:25,950 the number of heads obtained. 666 00:44:25,950 --> 00:44:29,440 So what kind of random variable is that? 667 00:44:29,440 --> 00:44:30,420 AUDIENCE: Bernoulli? 668 00:44:30,420 --> 00:44:30,910 AUDIENCE: Binomial. 669 00:44:30,910 --> 00:44:31,890 AUDIENCE: Bernoulli. 670 00:44:31,890 --> 00:44:32,380 AUDIENCE: Binomial. 671 00:44:32,380 --> 00:44:32,870 AUDIENCE: Binomial. 672 00:44:32,870 --> 00:44:33,780 VINA NGUYEN: Yes. 673 00:44:33,780 --> 00:44:34,994 Why? 674 00:44:34,994 --> 00:44:36,410 Because there's two tosses, right? 675 00:44:36,410 --> 00:44:38,490 It's not just one. 676 00:44:38,490 --> 00:44:40,370 AUDIENCE: Did you just make up 3/4? 677 00:44:40,370 --> 00:44:41,120 VINA NGUYEN: Yeah. 678 00:44:41,120 --> 00:44:42,010 Just make it up. 679 00:44:42,010 --> 00:44:43,384 AUDIENCE: Just wanted to be sure. 680 00:44:43,384 --> 00:44:44,210 VINA NGUYEN: Yeah. 681 00:44:44,210 --> 00:44:46,250 It could be biased. 682 00:44:46,250 --> 00:44:48,550 It was more interesting than 1/2. 683 00:44:48,550 --> 00:44:49,050 OK. 684 00:44:49,050 --> 00:44:54,300 So like I said, to describe a random variable 685 00:44:54,300 --> 00:44:55,410 you need probabilities. 686 00:44:55,410 --> 00:44:57,400 Otherwise, this doesn't matter. 687 00:44:57,400 --> 00:44:59,025 So what is your PMF? 688 00:45:10,214 --> 00:45:11,870 AUDIENCE: K could be 0. 689 00:45:25,916 --> 00:45:27,290 VINA NGUYEN: And like I said, you 690 00:45:27,290 --> 00:45:30,640 have to add the otherwise 0. 691 00:45:30,640 --> 00:45:32,240 So what is 0? 692 00:45:32,240 --> 00:45:34,170 The probability that your x equals 0? 693 00:45:38,571 --> 00:45:39,070 Anybody? 694 00:45:39,070 --> 00:45:41,180 AUDIENCE: [INAUDIBLE] 695 00:45:41,180 --> 00:45:42,751 VINA NGUYEN: Yeah. 696 00:45:42,751 --> 00:45:45,840 So we're just going to write that like this. 697 00:45:45,840 --> 00:45:46,340 1? 698 00:45:48,880 --> 00:45:50,883 Probability that you just get 1? 699 00:45:50,883 --> 00:45:51,466 AUDIENCE: 3/4. 700 00:45:55,230 --> 00:45:55,813 AUDIENCE: 3/4. 701 00:46:00,650 --> 00:46:04,490 VINA NGUYEN: But there's two different ways, right? 702 00:46:04,490 --> 00:46:07,280 So this is your tail-tail. 703 00:46:07,280 --> 00:46:08,461 This is your head-tail. 704 00:46:12,310 --> 00:46:13,099 And this one? 705 00:46:13,099 --> 00:46:13,682 AUDIENCE: 3/4. 706 00:46:20,556 --> 00:46:22,710 VINA NGUYEN: So the reason is 3/4, like you said. 707 00:46:22,710 --> 00:46:25,160 So we have two different probabilities for that. 708 00:46:25,160 --> 00:46:27,010 Otherwise, the 1/2 thing would throw us off. 709 00:46:30,960 --> 00:46:38,480 So expectation, like I've said, is your average. 710 00:46:38,480 --> 00:46:40,330 And the way we write it in probability 711 00:46:40,330 --> 00:46:46,300 is e of your random variable x. 712 00:46:46,300 --> 00:46:48,380 So x is your random variable. 713 00:46:48,380 --> 00:46:51,550 This means mean. 714 00:46:51,550 --> 00:46:55,600 So how would you figure out the mean of that? 715 00:46:55,600 --> 00:46:57,430 If we're just going to flip it twice, 716 00:46:57,430 --> 00:47:00,160 what is the average that we're expecting? 717 00:47:03,070 --> 00:47:08,890 AUDIENCE: [INAUDIBLE] 718 00:47:08,890 --> 00:47:10,224 AUDIENCE: Could you repeat that? 719 00:47:10,224 --> 00:47:10,889 VINA NGUYEN: Hm? 720 00:47:10,889 --> 00:47:12,790 AUDIENCE: Could you just repeat that again? 721 00:47:12,790 --> 00:47:13,706 VINA NGUYEN: Oh, yeah. 722 00:47:13,706 --> 00:47:15,590 So if you have your random variable and then 723 00:47:15,590 --> 00:47:17,700 probabilities of each event, how are you 724 00:47:17,700 --> 00:47:20,179 going to figure out the average? 725 00:47:20,179 --> 00:47:21,678 AUDIENCE: Take them all out and then 726 00:47:21,678 --> 00:47:24,330 divide by the number of them, right? 727 00:47:24,330 --> 00:47:26,490 VINA NGUYEN: Kind of. 728 00:47:26,490 --> 00:47:28,530 Not really. 729 00:47:28,530 --> 00:47:35,383 So let's say you have K equals 0 times the probability. 730 00:47:40,950 --> 00:47:46,610 So let's say that your x equals 0 times 731 00:47:46,610 --> 00:47:57,594 by this probability, which is plus 1 times that probability. 732 00:48:04,400 --> 00:48:07,010 So how do I finish this off? 733 00:48:07,010 --> 00:48:07,530 2, right? 734 00:48:07,530 --> 00:48:08,841 That's your last. 735 00:48:08,841 --> 00:48:11,787 AUDIENCE: 3/4 cubed times-- 736 00:48:16,220 --> 00:48:18,140 VINA NGUYEN: Does everyone see this? 737 00:48:18,140 --> 00:48:22,530 So the general way of saying that is this is sum-- 738 00:48:22,530 --> 00:48:25,160 I know someone asked this in one of the sheets-- 739 00:48:25,160 --> 00:48:29,150 sum of all your possible x's times 740 00:48:29,150 --> 00:48:34,460 the probability of that x, which is essentially 741 00:48:34,460 --> 00:48:36,020 what we just did. 742 00:48:36,020 --> 00:48:36,740 x equals 0. 743 00:48:36,740 --> 00:48:38,515 Probability of x equals 0. 744 00:48:38,515 --> 00:48:42,476 x equals 1, probability of x equals 1. 745 00:48:42,476 --> 00:48:44,630 x equals 2, probability of x equals 2. 746 00:48:47,456 --> 00:48:51,320 Everyone understand that formula? 747 00:48:51,320 --> 00:48:52,796 OK. 748 00:48:52,796 --> 00:48:55,082 AUDIENCE: [INAUDIBLE] 749 00:48:55,082 --> 00:48:55,748 VINA NGUYEN: Hm? 750 00:48:58,700 --> 00:48:59,410 AUDIENCE: Oh, OK. 751 00:48:59,410 --> 00:48:59,905 Got it. 752 00:48:59,905 --> 00:49:00,405 Yeah. 753 00:49:00,405 --> 00:49:01,885 Never mind. 754 00:49:01,885 --> 00:49:04,370 What comes out isn't a probability. 755 00:49:04,370 --> 00:49:05,120 VINA NGUYEN: Yeah. 756 00:49:05,120 --> 00:49:06,070 AUDIENCE: [INAUDIBLE] 757 00:49:06,070 --> 00:49:06,861 VINA NGUYEN: Right. 758 00:49:06,861 --> 00:49:11,931 So you have your real number line, 1, 2, 759 00:49:11,931 --> 00:49:14,180 and what you're expecting is that it falls right here. 760 00:49:23,110 --> 00:49:31,560 So this would be your [INAUDIBLE].. 761 00:49:31,560 --> 00:49:34,880 This is probably something like here. 762 00:49:34,880 --> 00:49:35,580 I don't know. 763 00:49:35,580 --> 00:49:38,040 You can graph it out later. 764 00:49:38,040 --> 00:49:43,760 Basically, your expected value is what the x 765 00:49:43,760 --> 00:49:47,625 is, not the probabilities. 766 00:49:47,625 --> 00:49:48,500 Does that make sense? 767 00:49:48,500 --> 00:49:49,083 Good question. 768 00:50:22,310 --> 00:50:25,380 Does everyone understand expectation? 769 00:50:25,380 --> 00:50:26,350 OK. 770 00:50:26,350 --> 00:50:31,100 So variance is basically your expectation of this. 771 00:50:31,100 --> 00:50:32,560 And what this is basically saying 772 00:50:32,560 --> 00:50:39,580 is the difference between your actual number 773 00:50:39,580 --> 00:50:41,650 and the mean squared. 774 00:50:41,650 --> 00:50:44,470 So a graphical way of looking at this-- 775 00:50:44,470 --> 00:50:45,950 AUDIENCE: Why do you square it? 776 00:50:45,950 --> 00:50:47,200 VINA NGUYEN: I'll get to that. 777 00:50:49,840 --> 00:50:51,700 This is not that graph. 778 00:50:51,700 --> 00:50:54,520 This is just like a random xy thing. 779 00:50:54,520 --> 00:51:03,350 So let's say you have a bunch of plots, whatever. 780 00:51:03,350 --> 00:51:05,170 So these are like your data points. 781 00:51:05,170 --> 00:51:09,504 And you figured out that this is the mean. 782 00:51:09,504 --> 00:51:10,960 So this is your expectation. 783 00:51:10,960 --> 00:51:14,710 This is what you expect to get. 784 00:51:14,710 --> 00:51:17,230 And x is each one of these. 785 00:51:17,230 --> 00:51:21,120 So this is like a real value of x. 786 00:51:21,120 --> 00:51:25,420 And x minus your mean would be this distance. 787 00:51:28,790 --> 00:51:31,170 So x minus the mean. 788 00:51:31,170 --> 00:51:32,790 x minus the mean. 789 00:51:32,790 --> 00:51:36,930 And the reason we square this is that for here, this 790 00:51:36,930 --> 00:51:38,500 might be a positive number. 791 00:51:38,500 --> 00:51:40,500 This might be a negative number. 792 00:51:40,500 --> 00:51:43,710 So we want to get rid of all this confusion 793 00:51:43,710 --> 00:51:45,330 and just square it. 794 00:51:45,330 --> 00:51:47,730 AUDIENCE: Doesn't that change the value? 795 00:51:47,730 --> 00:51:51,942 Couldn't you just do absolute values? 796 00:51:51,942 --> 00:51:52,900 VINA NGUYEN: You could. 797 00:51:52,900 --> 00:51:56,070 But to get the standard dev is a lot easier 798 00:51:56,070 --> 00:51:57,870 if you can see mathematically where 799 00:51:57,870 --> 00:52:00,330 the square root's coming from. 800 00:52:00,330 --> 00:52:05,370 So standard dev would be just to get rid of that square, 801 00:52:05,370 --> 00:52:06,750 and then you can get the actual. 802 00:52:06,750 --> 00:52:07,080 AUDIENCE: Right. 803 00:52:07,080 --> 00:52:08,079 VINA NGUYEN: Absolutely. 804 00:52:10,950 --> 00:52:13,930 So that's graphically what variance is. 805 00:52:13,930 --> 00:52:17,550 So if your data was really way off, 806 00:52:17,550 --> 00:52:19,636 then you get huge distances. 807 00:52:19,636 --> 00:52:21,760 And then you square it and you get a bigger number. 808 00:52:21,760 --> 00:52:24,270 So that shows that your data is more varied. 809 00:52:24,270 --> 00:52:27,240 And if your data is really tight, 810 00:52:27,240 --> 00:52:29,450 then your distance is small. 811 00:52:29,450 --> 00:52:29,950 Yeah. 812 00:52:29,950 --> 00:52:32,780 And you square that. 813 00:52:32,780 --> 00:52:33,446 OK. 814 00:52:33,446 --> 00:52:34,844 Does that make sense? 815 00:52:37,640 --> 00:52:41,600 So I haven't actually done this, but if you want you can-- 816 00:52:41,600 --> 00:52:47,730 if you want to calculate the variance of that 817 00:52:47,730 --> 00:52:52,400 you would just write out all of these using that PMF, 818 00:52:52,400 --> 00:52:54,950 because you have the mean. 819 00:52:54,950 --> 00:52:57,040 And just square everything. 820 00:52:57,040 --> 00:53:01,240 So because this expected value of this, 821 00:53:01,240 --> 00:53:08,356 your final formula is expected value of-- 822 00:53:15,796 --> 00:53:16,800 OK. 823 00:53:16,800 --> 00:53:20,820 So that's your final formula for that-- 824 00:53:20,820 --> 00:53:25,350 what you expect to get when you calculate all of these. 825 00:53:25,350 --> 00:53:27,060 Now remember that since you have-- 826 00:53:27,060 --> 00:53:30,570 this is a random variable, but you need to do it for x equals, 827 00:53:30,570 --> 00:53:34,548 in this case, 0, 1, 2, et cetera, et cetera. 828 00:53:42,810 --> 00:53:45,505 So I had problems-- 829 00:53:45,505 --> 00:53:46,150 oh, wait. 830 00:53:46,150 --> 00:53:48,820 Questions about any of this before I do the problems? 831 00:53:55,473 --> 00:53:57,437 AUDIENCE: That's a distinct, right? 832 00:53:57,437 --> 00:53:58,910 VINA NGUYEN: Hm? 833 00:53:58,910 --> 00:54:08,170 AUDIENCE: [INAUDIBLE] 834 00:54:08,170 --> 00:54:10,450 VINA NGUYEN: Oh, OK. 835 00:54:10,450 --> 00:54:13,690 This is just your mean. 836 00:54:13,690 --> 00:54:18,294 This is the expected value of all of these distances squared. 837 00:54:18,294 --> 00:54:19,460 So it's not graphed on here. 838 00:54:19,460 --> 00:54:21,581 It would be like a number that you would find. 839 00:54:21,581 --> 00:54:23,247 AUDIENCE: So you just add them together? 840 00:54:23,247 --> 00:54:26,490 [INAUDIBLE] 841 00:54:26,490 --> 00:54:27,240 VINA NGUYEN: Yeah. 842 00:54:27,240 --> 00:54:33,780 So like you would for 0 or something, you would do-- 843 00:54:38,630 --> 00:54:43,200 and your mean is 3/2. 844 00:54:43,200 --> 00:54:48,560 Square it, and then you have a probability that you would get. 845 00:54:56,352 --> 00:54:58,970 And we have those probabilities. 846 00:54:58,970 --> 00:55:03,560 Probability of whatever that was. 847 00:55:03,560 --> 00:55:06,530 And then you have to do the mean of this. 848 00:55:06,530 --> 00:55:09,140 So whatever you get for these, then you 849 00:55:09,140 --> 00:55:10,770 would times it by the probabilities 850 00:55:10,770 --> 00:55:13,140 and then get your expectation. 851 00:55:13,140 --> 00:55:15,200 So you're just converting your x's 852 00:55:15,200 --> 00:55:19,130 into this new random variable almost. 853 00:55:19,130 --> 00:55:24,140 So you can kind of think of this as like y or something. 854 00:55:24,140 --> 00:55:27,960 Or R, or K, whatever number you want. 855 00:55:27,960 --> 00:55:30,340 So then you would just use that same formula. 856 00:55:30,340 --> 00:55:32,126 OK? 857 00:55:32,126 --> 00:55:33,460 Any other questions? 858 00:55:36,450 --> 00:55:41,507 Did the solve last week go over the problems? 859 00:55:41,507 --> 00:55:42,590 Do you have any questions? 860 00:55:42,590 --> 00:55:44,488 You want me to go over any specific one? 861 00:55:44,488 --> 00:55:46,400 AUDIENCE: I didn't get the problems. 862 00:55:46,400 --> 00:55:47,941 VINA NGUYEN: Oh, you didn't get them? 863 00:55:49,970 --> 00:55:52,910 Does anyone have a copy she could borrow or share 864 00:55:52,910 --> 00:55:55,722 with because I only have one. 865 00:55:55,722 --> 00:55:56,222 Nobody? 866 00:56:02,080 --> 00:56:03,280 It's OK. 867 00:56:03,280 --> 00:56:05,167 I'll just borrow it if I need it. 868 00:56:05,167 --> 00:56:06,208 AUDIENCE: I can copy one. 869 00:56:06,208 --> 00:56:06,664 VINA NGUYEN: Hm? 870 00:56:06,664 --> 00:56:07,576 AUDIENCE: I can copy. 871 00:56:07,576 --> 00:56:09,400 VINA NGUYEN: It's fine. 872 00:56:09,400 --> 00:56:10,660 Only if people have questions. 873 00:56:10,660 --> 00:56:12,535 Does anyone have questions about any of them? 874 00:56:14,707 --> 00:56:16,290 Do you want me to go over all of them? 875 00:56:19,713 --> 00:56:33,349 AUDIENCE: [INAUDIBLE] 876 00:56:33,349 --> 00:56:34,890 VINA NGUYEN: What was the fourth one? 877 00:56:34,890 --> 00:56:37,414 AUDIENCE: Something to do with a deck of cards. 878 00:56:37,414 --> 00:56:38,205 VINA NGUYEN: Sorry. 879 00:56:38,205 --> 00:56:40,070 What was the fourth one? 880 00:56:40,070 --> 00:56:40,880 OK. 881 00:56:40,880 --> 00:56:41,950 Do you want me to go over that one? 882 00:56:41,950 --> 00:56:42,449 AUDIENCE: Yeah. 883 00:56:42,449 --> 00:56:43,115 VINA NGUYEN: OK. 884 00:56:46,940 --> 00:56:49,450 Can someone read it out because I don't have it. 885 00:56:49,450 --> 00:56:52,384 AUDIENCE: Oh, basically, you had a [INAUDIBLE] 886 00:56:52,384 --> 00:56:54,340 cards into four hands. 887 00:56:54,340 --> 00:56:56,510 What's the probability of one ace in each hand? 888 00:56:56,510 --> 00:56:57,622 VINA NGUYEN: OK. 889 00:56:57,622 --> 00:56:59,730 Does anyone need this? 890 00:56:59,730 --> 00:57:00,230 No. 891 00:57:00,230 --> 00:57:01,760 Good. 892 00:57:01,760 --> 00:57:04,261 I can say it again. 893 00:57:04,261 --> 00:57:04,760 This? 894 00:57:25,320 --> 00:57:28,490 So this was the last problem you did? 895 00:57:28,490 --> 00:57:31,850 So 52 cards. 896 00:57:31,850 --> 00:57:33,830 How many hands? 897 00:57:33,830 --> 00:57:35,400 How many hands was it? 898 00:57:35,400 --> 00:57:36,010 Four hands. 899 00:57:39,940 --> 00:57:41,525 And one ace per hand. 900 00:57:50,970 --> 00:57:54,019 So what was the main question, just how to do it, or-- 901 00:57:54,019 --> 00:57:54,560 AUDIENCE: No. 902 00:57:54,560 --> 00:57:56,726 Just what's the probability of one ace in each hand. 903 00:57:56,726 --> 00:57:58,800 VINA NGUYEN: Oh, OK. 904 00:57:58,800 --> 00:58:02,400 So there are two ways, like we mentioned before. 905 00:58:04,980 --> 00:58:12,750 The easier way, in my opinion, is the sequential way. 906 00:58:12,750 --> 00:58:15,210 So let me read. 907 00:58:15,210 --> 00:58:15,710 OK. 908 00:58:18,270 --> 00:58:20,190 All right. 909 00:58:20,190 --> 00:58:24,870 So the first hand, we have 13 slots. 910 00:58:24,870 --> 00:58:27,360 13 slots. 911 00:58:27,360 --> 00:58:29,870 Second hand has that many. 912 00:58:35,730 --> 00:58:39,740 So if you were to put one ace here, 913 00:58:39,740 --> 00:58:45,750 that's just 52 different possibilities 914 00:58:45,750 --> 00:58:47,760 over the total number. 915 00:58:47,760 --> 00:58:48,950 So we can go to either one. 916 00:58:48,950 --> 00:58:49,500 Any one. 917 00:58:49,500 --> 00:58:50,541 It doesn't really matter. 918 00:58:50,541 --> 00:58:52,050 AUDIENCE: Oh, right. 919 00:58:52,050 --> 00:58:56,160 VINA NGUYEN: If you put one here, how do we calculate that? 920 00:58:56,160 --> 00:58:58,870 How many spots are left? 921 00:58:58,870 --> 00:58:59,533 51, right? 922 00:58:59,533 --> 00:59:00,074 AUDIENCE: 51. 923 00:59:00,074 --> 00:59:01,930 But you can only go to 39. 924 00:59:01,930 --> 00:59:03,366 VINA NGUYEN: Yeah. 925 00:59:03,366 --> 00:59:05,340 Does everyone understand that? 926 00:59:05,340 --> 00:59:08,460 So you take out all of these as your possible choices, 927 00:59:08,460 --> 00:59:10,920 but they are still possible choices, but just not 928 00:59:10,920 --> 00:59:14,785 where the ace can go. 929 00:59:14,785 --> 00:59:17,240 AUDIENCE: Then 26 out of 50. 930 00:59:21,659 --> 00:59:24,120 VINA NGUYEN: Mhm. 931 00:59:24,120 --> 00:59:26,430 Does everyone understand that? 932 00:59:26,430 --> 00:59:31,804 First ace, second ace, third ace, fourth ace. 933 00:59:31,804 --> 00:59:37,060 So that's the intuitive way, in my opinion. 934 00:59:37,060 --> 00:59:41,382 The second way we can use, counting. 935 00:59:41,382 --> 00:59:42,840 We have partitions, because there's 936 00:59:42,840 --> 00:59:44,130 four different partitions. 937 00:59:44,130 --> 00:59:44,920 Stuff like that. 938 01:00:02,490 --> 01:00:07,860 So you have your top part and your bottom part. 939 01:00:07,860 --> 01:00:12,510 So your bottom part is the total number of combinations. 940 01:00:15,180 --> 01:00:18,615 And this is only the ones that match your criteria. 941 01:00:22,410 --> 01:00:24,690 So we'll do this one first. 942 01:00:24,690 --> 01:00:28,460 Since we know partitions, we have the bottom part, 943 01:00:28,460 --> 01:00:33,251 which is the denominator. 944 01:00:37,400 --> 01:00:39,660 Do you guys remember partitions? 945 01:00:39,660 --> 01:00:41,171 Kind of? 946 01:00:41,171 --> 01:00:41,670 Kind of? 947 01:00:41,670 --> 01:00:42,570 OK. 948 01:00:42,570 --> 01:00:46,320 So there should be two spots. 949 01:00:46,320 --> 01:00:50,580 And we have four hands, and we have 13 slots 950 01:00:50,580 --> 01:00:52,230 in each of those hands. 951 01:00:52,230 --> 01:00:56,556 So you partition that like this. 952 01:00:56,556 --> 01:00:59,510 AUDIENCE: Oh, [INAUDIBLE] 953 01:00:59,510 --> 01:01:01,380 VINA NGUYEN: OK. 954 01:01:01,380 --> 01:01:03,690 So this is the number of combinations you can have. 955 01:01:03,690 --> 01:01:06,960 13 cards in four different ways. 956 01:01:06,960 --> 01:01:07,460 52. 957 01:01:16,860 --> 01:01:20,010 First thing you want to count is how many different ways 958 01:01:20,010 --> 01:01:22,260 can you place those aces. 959 01:01:22,260 --> 01:01:29,730 And that's four ways because you ace one, ace two, ace three, 960 01:01:29,730 --> 01:01:31,420 ace four. 961 01:01:31,420 --> 01:01:35,230 You have four selections for your first slot. 962 01:01:35,230 --> 01:01:38,410 Then, once you put it in, you have 3, 2, 1. 963 01:01:38,410 --> 01:01:40,890 So that's four. 964 01:01:40,890 --> 01:01:42,910 And then now you need to do this kind of thing. 965 01:01:42,910 --> 01:01:47,560 So what's your remaining partition left? 966 01:01:47,560 --> 01:01:50,560 52 minus 4 cards is 48. 967 01:01:50,560 --> 01:01:53,290 So we only have this many left. 968 01:01:53,290 --> 01:01:55,690 And then the remaining slots? 969 01:01:55,690 --> 01:02:02,214 AUDIENCE: [INAUDIBLE] 970 01:02:02,214 --> 01:02:02,880 VINA NGUYEN: OK. 971 01:02:02,880 --> 01:02:05,000 Oh, it was kind of written differently. 972 01:02:05,000 --> 01:02:08,020 But basically, this goes on top, that goes on button. 973 01:02:08,020 --> 01:02:11,590 So you calculate it out and it looks like that.