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.