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.