In the previous two lectures we talked about multicriterion decision making and then spatial decision making. Where we want to go next is decision making under uncertainty where there are some probabilities involved. So to do that first I want to take a little time out and talk for a moment about probability. Now if you've already taken a class in probability or if you know a lot of probability, you can skip this little unit. If you haven't, this'll give you enough understanding that you can, you know, need what you, you'll know what you need to know, to do what we're gonna do with respects to decision making under uncertainty. So probability. Our probabilities are just the odds that something happens. And so when you break down probabilities they have to satisfy three axioms. First axiom is that any probability is between zero and one. So if something can happen, it's probably zero. If something's definitely going to happen, it's probably one. Now even if you're 100, you're totally sure something's gonna happen, the probability can't be bigger than one. So you can't say, I think there's a 110 percent chance this is gonna [laugh] happen. No, it's [laugh] gotta be between zero percent and 100%. Second actions are more complicated. You have to make a distinction between outcomes and events. So an outcome is just any individual thing that can happen. An event is a subset of outcomes. So if I wrote down all possible outcomes, then the sum. Of those probabilities, has to equal one. So if I think about flipping a coin, right, there's two outcomes, heads or tails. The probability of heads is a half, the probability of tails is a half. And when I sum those two things together, I get one, that's the second axiom, easy. Third axion. If I have an event, known event would be a set of outcomes. And the event a is contained in the event b, then the probability of a is less than the probability of b. So, one event might be. That I get [inaudible] little sets. Another event might be that I get a head or a tails. But the probability of getting a head is a half. The probability of getting a head or a tails is one. And since getting a head is subset, right? Of getting a head or a tails. The probability of a head, one-half, is less than the probability of getting a head or a tails, which is one, that's the third axiom. So that's it, those are the three things. Probability of any outcome or event is between zero and one, could be zero, could be one, but it's somewhere in that range. That, if I add up the probabilities of all the different outcomes, those [inaudible] sum up to one. And then if I have one event that's a subset of another event, this is this axion, then that first event has a smaller probability than the second event. So that's the axioms. So there's actually three different types of probabilities. The first type of probability are classical probabilities. So these are the sort of things that mathematicians play with when you think about things like dice and roulette wheels and things like that. So for example, if I roll a die, I can sort of logically or classically assume that the probability of, you know, getting a four would be just one-sixth, and the probability of getting an even number would be one-half, and the probability of getting an odd number would be one-half. So this is classical probability where you can sort of write down mathematically in some pure sense what each probability would be. And there's a second type of probability, which is frequency. So here like with a, with a die we know that it's gonna be a six because the die is sort of equally shaped. There's gonna be other things where we don't know but what we can do is we can count. We can sort of do a frequency count. So we've got lots of data and we can look at all that data and from that data we can, you know, make an estimate of what we think the probability is. So for example suppose I ask you the following question. Do more words being with R? >> Or the more words have R as their third level, le, letter, right? So distinctive question. Now, what you could do is you can just guess. Right? Give that, well I'm guessing that two percent of words have Rs their third le, letter and eight percent of words begin with R. Another thing you could do is you could open up the dictionary and you could count, right? So you could just, first of all, you could just sort of look at how many pages are there that seem to begin with R and you could maybe get that, you know, six percent or something begin with R. And then you could randomly look through the dictionary looking at words and see what percentage of words have R their third letter and you might find that, that may be like eleven percent or something. And you might find out, oh my goodness, that this is actually bigger. Well, what' you're doing is you're sort of estimating through frequency what a probability of having R as it's third letter is and estimating through frequency what the probability of having R as it's first letter is. So, frequency just means you count things, right, and then you figure out the probability from there. So it's not a pure probability like rolling a die that it's one-sixth but it's just how often it seems to happen. So if you look at something like, is it gonna rain next July seventh, or June seventh, I'm sorry. What you can do is you can go back and look over the last hundred years. And you could take 100 years of data, and on 26 of those days. It's rained, and on 74 of those days, it hasn't rained, and so then you can say I think the probability of rain is 26%. Now again this isn't like rolling a die. This is just counting it up, right, and this is a frequency estimate of what the probability is. When you make these frequency estimates, you're making some strong assumptions, one of which is that what we call stationarity, that nothing has changed over the last hundred years, that the probability of rain has been stationary and hasn't changed and so this is a good predictor. So, ideally, right, we know classically, the probability of something. And if we don't know classically, then the next best thing would be to use all that data we've got out there in the world, and do a frequent list account. Sometimes, we can't do either one of those things, and we're stuck with subjective probabilities. So these are cases where we kinda have to guess, or have to, and, we'll talk about this, actually. What we wanna do is use a model, right? We wanna have some sort of model we could use to figure out how, what a subjective probability is. So, for example, here's, A case that is sometimes given by psychologists. So Shelly majored in political science and was very involved in college Republicans. Write down probabilities for the following events. So I've got, let's think. Now [inaudible] think Shelley's a political scientist, right? That's sort of interesting. She's a republican. So that means [inaudible] conservative political scientist. You know, maybe she's, you know munched in money. What are the probabilities she'd do these things? Well, flights attendant, I might think, well boy, that's not very likely, right? Five%. Blogger, I could think, you know, maybe blogging, maybe there's a ten percent chance she blogs. Because, you know, she was a political science major and she was a republican. So maybe she likes to blog. Flight attending while finishing your MBA. Well that seems actually pretty reasonable. Let's give that a ten percent chance. And then medical field, let's say, you know, medical, lot of people in the medical field, so let's just put a fifteen percent chance she's working in the medical field because that's about what, you know, the base rate for what people work in the medical field. So these would be my probability estimates [inaudible] subjectively writing those things down. Well, let's look at these a little more carefully. I did something wrong. What did I do wrong? Remember our three axioms. What were our three axioms? Axiom one was, right, that the probabilities had to be between zero and one. Right? Axiom two was that all probabilities summed up to one, so the sum was one. And the third one had to do with event A, was contained in event B. So let's go back and look at. What I did. What did I do? I assumed event A, that she was a flight attendant, that this was only true five percent of the time. And event C, [inaudible] she was a flight attendant while finishing her MBA was ten%. Well this can't be, right? Because if she's a flight attendant. Right? That's this event, event A, right, contains event C. So if she's a flight attendant while finishing her MBA then she's also a flight attendant. So this number, right, this ten percent has to be smaller than the five%. So we made a mistake. And in fact, this example, the one I gave, remember I said psychologists like to use this, this is an example where we see a bias, where people make mistakes. And in a way we'll actually talk about these sort of biases. So subjective probabilities are dangerous, because when you start writing down numbers, right? We may not satisfy those axioms. And so then our probabilities don't make any sense. So, suppose someone asks you a question like, will housing prices will go up next year. How do you do it? Well. One thing you could do is just guess. Maybe we model think of the direction the housing is moving and ten from there make some sort of assessment whether housing prices will go up. So when we think about cases where we don't have a classical probability, right, when you pick up a probability, probability textbook they'll say really there's only, there's two things you can do. One is you can do a frequency method and the other is you can use subjective, subjective method. We're actually going to argue for a third way which is even though these probabilities are subjective, you want to think of these as sort of model-based probabilities. What we're going to do is try to construct a model and based on that model figure out what we think the probability of an event will be. Here we have probability in a nut shell, right. There's three actions. Probability's there between zero and one. Probability of E, if you add up all the possible outcomes, that it's the sum to one. And if one event contains another event, it's gotta be more likely. That's all, that's it, those three Xes. And again, there's three types of probability. One is classical, where we know, sort of mathematically, why a probability is what it is. Second is frequency based where we've got all sorts of data and based on that data then we know, we have some good estimate of what we think the probability is. Third case is what's often called subjectives, what we don't have data and we don't have a class co reason, so we sorta gotta, we gotta guess, and rather than guess what we can use is try and gather certain model and use the model to get a sum, you know estimate of what we think the probability is going to be. And so these probabilities then are gonna come into play. In the next lecture we think about. How do we make choices when we don't know something for sure? We know that there is some probability of it raining or some probability of prices going up. So, that's we're moving next. Decision making where we've got uncertainty in these probabilities. Thank you.