1 00:00:00,000 --> 00:00:03,806 In the previous two lectures we talked about multicriterion decision making and 2 00:00:03,806 --> 00:00:07,564 then spatial decision making. Where we want to go next is decision making under 3 00:00:07,564 --> 00:00:10,895 uncertainty where there are some probabilities involved. So to do that 4 00:00:10,895 --> 00:00:14,844 first I want to take a little time out and talk for a moment about probability. Now 5 00:00:14,844 --> 00:00:18,222 if you've already taken a class in probability or if you know a lot of 6 00:00:18,222 --> 00:00:21,743 probability, you can skip this little unit. If you haven't, this'll give you 7 00:00:21,743 --> 00:00:25,454 enough understanding that you can, you know, need what you, you'll know what you 8 00:00:25,454 --> 00:00:29,165 need to know, to do what we're gonna do with respects to decision making under 9 00:00:29,165 --> 00:00:32,597 uncertainty. So probability. Our probabilities are just the odds that 10 00:00:32,597 --> 00:00:36,788 something happens. And so when you break down probabilities they have to satisfy 11 00:00:36,788 --> 00:00:41,036 three axioms. First axiom is that any probability is between zero and one. So if 12 00:00:41,036 --> 00:00:45,467 something can happen, it's probably zero. If something's definitely going to happen, 13 00:00:45,467 --> 00:00:50,122 it's probably one. Now even if you're 100, you're totally sure something's gonna 14 00:00:50,122 --> 00:00:54,528 happen, the probability can't be bigger than one. So you can't say, I think 15 00:00:54,528 --> 00:00:59,055 there's a 110 percent chance this is gonna [laugh] happen. No, it's [laugh] gotta be 16 00:00:59,055 --> 00:01:03,220 between zero percent and 100%. Second actions are more complicated. You have to 17 00:01:03,220 --> 00:01:07,626 make a distinction between outcomes and events. So an outcome is just any 18 00:01:07,626 --> 00:01:12,455 individual thing that can happen. An event is a subset of outcomes. So if I wrote 19 00:01:12,455 --> 00:01:17,528 down all possible outcomes, then the sum. Of those probabilities, has to equal one. 20 00:01:17,528 --> 00:01:22,314 So if I think about flipping a coin, right, there's two outcomes, heads or 21 00:01:22,314 --> 00:01:27,631 tails. The probability of heads is a half, the probability of tails is a half. And 22 00:01:27,631 --> 00:01:32,682 when I sum those two things together, I get one, that's the second axiom, easy. 23 00:01:32,682 --> 00:01:38,039 Third axion. If I have an event, known event would be a set of outcomes. And the 24 00:01:38,039 --> 00:01:43,496 event a is contained in the event b, then the probability of a is less than the 25 00:01:43,496 --> 00:01:48,673 probability of b. So, one event might be. That I get [inaudible] little sets. 26 00:01:48,673 --> 00:01:54,563 Another event might be that I get a head or a tails. But the probability of getting 27 00:01:54,563 --> 00:01:59,744 a head is a half. The probability of getting a head or a tails is one. And 28 00:01:59,744 --> 00:02:04,405 since getting a head is subset, right? Of getting a head or a tails. The probability 29 00:02:04,405 --> 00:02:07,887 of a head, one-half, is less than the probability of getting a head or a tails, 30 00:02:07,887 --> 00:02:11,325 which is one, that's the third axiom. So that's it, those are the three things. 31 00:02:11,325 --> 00:02:14,943 Probability of any outcome or event is between zero and one, could be zero, could 32 00:02:14,943 --> 00:02:18,652 be one, but it's somewhere in that range. That, if I add up the probabilities of all 33 00:02:18,652 --> 00:02:22,274 the different outcomes, those [inaudible] sum up to one. And then if I have one 34 00:02:22,274 --> 00:02:26,221 event that's a subset of another event, this is this axion, then that first event 35 00:02:26,221 --> 00:02:30,447 has a smaller probability than the second event. So that's the axioms. So there's 36 00:02:30,447 --> 00:02:34,277 actually three different types of probabilities. The first type of 37 00:02:34,277 --> 00:02:38,745 probability are classical probabilities. So these are the sort of things that 38 00:02:38,745 --> 00:02:43,504 mathematicians play with when you think about things like dice and roulette wheels 39 00:02:43,504 --> 00:02:48,088 and things like that. So for example, if I roll a die, I can sort of logically or 40 00:02:48,088 --> 00:02:52,789 classically assume that the probability of, you know, getting a four would be just 41 00:02:52,789 --> 00:02:57,548 one-sixth, and the probability of getting an even number would be one-half, and the 42 00:02:57,548 --> 00:03:01,860 probability of getting an odd number would be one-half. So this is classical 43 00:03:01,860 --> 00:03:06,075 probability where you can sort of write down mathematically in some pure sense 44 00:03:06,075 --> 00:03:09,917 what each probability would be. And there's a second type of probability, 45 00:03:09,917 --> 00:03:14,078 which is frequency. So here like with a, with a die we know that it's gonna be a 46 00:03:14,078 --> 00:03:18,133 six because the die is sort of equally shaped. There's gonna be other things 47 00:03:18,133 --> 00:03:22,135 where we don't know but what we can do is we can count. We can sort of do a 48 00:03:22,135 --> 00:03:26,617 frequency count. So we've got lots of data and we can look at all that data and from 49 00:03:26,617 --> 00:03:30,667 that data we can, you know, make an estimate of what we think the probability 50 00:03:30,672 --> 00:03:35,649 is. So for example suppose I ask you the following question. Do more words being 51 00:03:35,649 --> 00:03:40,594 with R? >> Or the more words have R as their third level, le, letter, right? So 52 00:03:40,594 --> 00:03:45,272 distinctive question. Now, what you could do is you can just guess. Right? Give 53 00:03:45,272 --> 00:03:48,904 that, well I'm guessing that two percent of words have Rs their third le, letter 54 00:03:48,904 --> 00:03:52,586 and eight percent of words begin with R. Another thing you could do is you could 55 00:03:52,586 --> 00:03:56,474 open up the dictionary and you could count, right? So you could just, first of 56 00:03:56,474 --> 00:04:00,515 all, you could just sort of look at how many pages are there that seem to begin 57 00:04:00,515 --> 00:04:04,248 with R and you could maybe get that, you know, six percent or something begin with 58 00:04:04,248 --> 00:04:08,085 R. And then you could randomly look through the dictionary looking at words 59 00:04:08,085 --> 00:04:12,074 and see what percentage of words have R their third letter and you might find 60 00:04:12,074 --> 00:04:15,706 that, that may be like eleven percent or something. And you might find out, oh my 61 00:04:15,706 --> 00:04:19,635 goodness, that this is actually bigger. Well, what' you're doing is you're sort of 62 00:04:19,635 --> 00:04:23,134 estimating through frequency what a probability of having R as it's third 63 00:04:23,134 --> 00:04:27,106 letter is and estimating through frequency what the probability of having R as it's 64 00:04:27,106 --> 00:04:30,700 first letter is. So, frequency just means you count things, right, and then you 65 00:04:30,700 --> 00:04:34,672 figure out the probability from there. So it's not a pure probability like rolling a 66 00:04:34,672 --> 00:04:38,638 die that it's one-sixth but it's just how often it seems to happen. So if you look 67 00:04:38,638 --> 00:04:42,842 at something like, is it gonna rain next July seventh, or June seventh, I'm sorry. 68 00:04:42,842 --> 00:04:47,099 What you can do is you can go back and look over the last hundred years. And you 69 00:04:47,099 --> 00:04:51,285 could take 100 years of data, and on 26 of those days. It's rained, and on 74 of 70 00:04:51,285 --> 00:04:55,599 those days, it hasn't rained, and so then you can say I think the probability of 71 00:04:55,599 --> 00:04:59,749 rain is 26%. Now again this isn't like rolling a die. This is just counting it 72 00:04:59,749 --> 00:05:03,791 up, right, and this is a frequency estimate of what the probability is. When 73 00:05:03,791 --> 00:05:08,160 you make these frequency estimates, you're making some strong assumptions, one of 74 00:05:08,160 --> 00:05:12,474 which is that what we call stationarity, that nothing has changed over the last 75 00:05:12,474 --> 00:05:16,515 hundred years, that the probability of rain has been stationary and hasn't 76 00:05:16,515 --> 00:05:20,722 changed and so this is a good predictor. So, ideally, right, we know classically, 77 00:05:20,722 --> 00:05:25,151 the probability of something. And if we don't know classically, then the next best 78 00:05:25,151 --> 00:05:29,416 thing would be to use all that data we've got out there in the world, and do a 79 00:05:29,416 --> 00:05:33,844 frequent list account. Sometimes, we can't do either one of those things, and we're 80 00:05:33,844 --> 00:05:38,109 stuck with subjective probabilities. So these are cases where we kinda have to 81 00:05:38,109 --> 00:05:42,373 guess, or have to, and, we'll talk about this, actually. What we wanna do is use a 82 00:05:42,373 --> 00:05:46,802 model, right? We wanna have some sort of model we could use to figure out how, what 83 00:05:46,802 --> 00:05:51,009 a subjective probability is. So, for example, here's, A case that is sometimes 84 00:05:51,009 --> 00:05:55,411 given by psychologists. So Shelly majored in political science and was very involved 85 00:05:55,411 --> 00:05:59,393 in college Republicans. Write down probabilities for the following events. So 86 00:05:59,393 --> 00:06:03,322 I've got, let's think. Now [inaudible] think Shelley's a political scientist, 87 00:06:03,322 --> 00:06:07,409 right? That's sort of interesting. She's a republican. So that means [inaudible] 88 00:06:07,409 --> 00:06:11,286 conservative political scientist. You know, maybe she's, you know munched in 89 00:06:11,286 --> 00:06:15,478 money. What are the probabilities she'd do these things? Well, flights attendant, I 90 00:06:15,478 --> 00:06:19,195 might think, well boy, that's not very likely, right? Five%. Blogger, I could 91 00:06:19,195 --> 00:06:22,692 think, you know, maybe blogging, maybe there's a ten percent chance she blogs. 92 00:06:22,692 --> 00:06:26,817 Because, you know, she was a political science major and she was a republican. So 93 00:06:26,817 --> 00:06:30,681 maybe she likes to blog. Flight attending while finishing your MBA. Well that seems 94 00:06:30,681 --> 00:06:33,607 actually pretty reasonable. Let's give that a ten percent chance. And then 95 00:06:33,607 --> 00:06:36,759 medical field, let's say, you know, medical, lot of people in the medical 96 00:06:36,759 --> 00:06:40,180 field, so let's just put a fifteen percent chance she's working in the medical field 97 00:06:40,180 --> 00:06:43,647 because that's about what, you know, the base rate for what people work in the 98 00:06:43,647 --> 00:06:46,754 medical field. So these would be my probability estimates [inaudible] 99 00:06:46,754 --> 00:06:50,220 subjectively writing those things down. Well, let's look at these a little more 100 00:06:50,220 --> 00:06:54,371 carefully. I did something wrong. What did I do wrong? Remember our three axioms. 101 00:06:54,371 --> 00:07:00,128 What were our three axioms? Axiom one was, right, that the probabilities had to be 102 00:07:00,128 --> 00:07:06,019 between zero and one. Right? Axiom two was that all probabilities summed up to one, 103 00:07:06,019 --> 00:07:12,077 so the sum was one. And the third one had to do with event A, was contained in event 104 00:07:12,077 --> 00:07:17,434 B. So let's go back and look at. What I did. What did I do? I assumed event A, 105 00:07:17,434 --> 00:07:22,687 that she was a flight attendant, that this was only true five percent of the time. 106 00:07:22,687 --> 00:07:28,372 And event C, [inaudible] she was a flight attendant while finishing her MBA was 107 00:07:28,372 --> 00:07:32,938 ten%. Well this can't be, right? Because if she's a flight attendant. Right? That's 108 00:07:32,938 --> 00:07:37,240 this event, event A, right, contains event C. So if she's a flight attendant while 109 00:07:37,240 --> 00:07:41,761 finishing her MBA then she's also a flight attendant. So this number, right, this ten 110 00:07:41,761 --> 00:07:45,574 percent has to be smaller than the five%. So we made a mistake. And in fact, this 111 00:07:45,574 --> 00:07:49,986 example, the one I gave, remember I said psychologists like to use this, this is an 112 00:07:49,986 --> 00:07:54,507 example where we see a bias, where people make mistakes. And in a way we'll actually 113 00:07:54,507 --> 00:07:58,959 talk about these sort of biases. So subjective probabilities are dangerous, 114 00:07:58,959 --> 00:08:03,798 because when you start writing down numbers, right? We may not satisfy those 115 00:08:03,798 --> 00:08:09,546 axioms. And so then our probabilities don't make any sense. So, suppose someone 116 00:08:09,546 --> 00:08:17,997 asks you a question like, will housing prices will go up next year. How do you do 117 00:08:17,997 --> 00:08:25,708 it? Well. One thing you could do is just guess. Maybe we model think of the 118 00:08:25,708 --> 00:08:34,158 direction the housing is moving and ten from there make some sort of assessment 119 00:08:34,158 --> 00:08:39,950 whether housing prices will go up. So when we think about cases where we don't have a 120 00:08:39,950 --> 00:08:43,013 classical probability, right, when you pick up a probability, probability 121 00:08:43,013 --> 00:08:46,551 textbook they'll say really there's only, there's two things you can do. One is you 122 00:08:46,551 --> 00:08:50,176 can do a frequency method and the other is you can use subjective, subjective method. 123 00:08:50,176 --> 00:08:53,325 We're actually going to argue for a third way which is even though these 124 00:08:53,325 --> 00:08:56,734 probabilities are subjective, you want to think of these as sort of model-based 125 00:08:56,734 --> 00:09:00,315 probabilities. What we're going to do is try to construct a model and based on that 126 00:09:00,315 --> 00:09:03,551 model figure out what we think the probability of an event will be. Here we 127 00:09:03,551 --> 00:09:07,156 have probability in a nut shell, right. There's three actions. Probability's there 128 00:09:07,156 --> 00:09:11,373 between zero and one. Probability of E, if you add up all the possible outcomes, that 129 00:09:11,373 --> 00:09:15,387 it's the sum to one. And if one event contains another event, it's gotta be more 130 00:09:15,387 --> 00:09:19,300 likely. That's all, that's it, those three Xes. And again, there's three types of 131 00:09:19,300 --> 00:09:22,958 probability. One is classical, where we know, sort of mathematically, why a 132 00:09:22,958 --> 00:09:26,834 probability is what it is. Second is frequency based where we've got all sorts 133 00:09:26,834 --> 00:09:30,714 of data and based on that data then we know, we have some good estimate of what 134 00:09:30,714 --> 00:09:35,042 we think the probability is. Third case is what's often called subjectives, what we 135 00:09:35,042 --> 00:09:39,361 don't have data and we don't have a class co reason, so we sorta gotta, we gotta 136 00:09:39,361 --> 00:09:43,735 guess, and rather than guess what we can use is try and gather certain model and 137 00:09:43,735 --> 00:09:47,999 use the model to get a sum, you know estimate of what we think the probability 138 00:09:47,999 --> 00:09:52,592 is going to be. And so these probabilities then are gonna come into play. In the next 139 00:09:52,592 --> 00:09:56,552 lecture we think about. How do we make choices when we don't know something for 140 00:09:56,552 --> 00:09:59,738 sure? We know that there is some probability of it raining or some 141 00:09:59,738 --> 00:10:03,647 probability of prices going up. So, that's we're moving next. Decision making where 142 00:10:03,647 --> 00:10:06,302 we've got uncertainty in these probabilities. Thank you.