We're working in supplement C, on the problems, this is our first problem and we're
working on slide seven and eight in supplement C. We're gonna be looking at
drawing a decision tree.
A decision tree is a graphical model used to determine our
estimated or expected monetary value.
It..
uses squares or rectangles as our decision points, circles as an
event node, and a branch or a straight line where we're going to
write our probability.
These symbols are very important because it actually tells us what there is
uh... what each thing is and they're identifiers... so if we take a look at
slide eight.
We're looking at the problem.
uh... WGIV; Giveaway 93; this is a local radio station, currently has a
giveaway program,
giveaway contest underway.
Once a contestant is selected, he or she is given two choices... take ninety three
dollars in cash...
or pick one of three envelopes.
The contents of those envelopes are five hundred dollars,
one hundred dollars, or merchandise which is estimated to be about twenty dollars,
which I believe is probably crap!
We want to draw decision tree for this decision and then decide what
decision
should be made to achieve the expected monetary value.
What would you do before we even draw the decision tree? Would you take
the envelope,
or would you take the straight cash?
I'm one of those people who is very low risk,
I would probably take the envelope!
Let's take a look now
at that decision.
What do we need to do?
We're gonna start by drawing our decision tree, and our decision point we
said
was
a square,
or a rectangle. Now you don't have to write anything in there, but I'm the type
of person who needs as much help as possible. So...
I want to remember what I'm looking at, so I know it's going to be cash
or an envelope.
So I'm looking
at my decision points, do I want to take cash or do I want to take an envelope? So
those are my two events that can happen.
I can take
cash
or I can take an envelope.
So we're getting our decision tree built here
and I know that, if I do take the cash,
it is a probability of 100%
that I'm going to receive
ninety three dollars.
Notice at the branch
is the line
with the probability written above it and the outcome will be
the ninety three dollars!
Now, the envelope...
I have three possible outcomes
if this takes place.
I might receive
five hundred dollars,
I might receive
one hundred dollars, and I can get that envelope with, you know, the coupons and
everything else
that's worth about twenty bucks.
Now because we want to get as close as possible we're not gonna use
decimal points here. We will be working with fractions and we have a
one-in-three chance
of getting five hundred dollars,
a one-in-three chance of getting one hundred dollars,
and a one-in-three chance of getting twenty
dollars.
This right here
is your decision tree!
There's nothing more to it! You have
your decision point,
your event nodes,
and then the possible outcomes,
dealing also with
our probabilities!
So, using this decision tree we are now going to
do our expected monetary values.
uh... Expected monetary value! Let's start with our expected
monetary value
for cash!
What was
the probability?
Probability
was one point zero (1.0).
So I have a 1.0 chance chance, or a hundred percent chance, of getting
ninety three dollars!
Do the math,
I can actually do that one in my head,
and it is ninety three dollars!
The expected
monetary value, if I take an envelope,
is a little bit harder.
If you take a look, I have three different options
that can happen, or three different outcomes that might occur.
I have a one-in-three chance of getting the five hundred dollars, a one-in-three
chance of getting one hundred dollars,
and a one-in-three chance of getting twenty dollars!
So, if we take a look at this,
my expected monetary value is a one-in-three chance
of getting five hundred dollars
plus
a one-in-three chance
of getting
one hundred dollars
plus a one-in-three chance
of getting twenty dollars.
Now, if we take a look
back here for just a minute
at the definition
of expected monetary value
which is Slide 5 in Supplement C,
it is found by summing,
for every state that can occur, the products obtained
when the probability that the state will occur
is multiplied by the amount that is expected to be received
if the act as taken, and that state occurs!
So if we take a look,
we have the probability
times the outcome if that occurs, so it's a one-in-three chance times that,
summing up each and every single one of those probabilities times that
possible outcome.
We're then going to add them up so doing a little bit of math here,
one-third
times
five hundred... which is 1 times 500 divided by 3
and this is looking at a monetary value so we only have to take it out to two
decimal points,
is one hundred and sixty six point
six seven (166.67).
One third time's one hundred or 1 times 100 divided by 3
is 33
point 33 (33.33)
and one third times 20 or 1
times 20 then divide it by 3 is 6
point 67 (6.67).
If we then take all of these and add them together
we should have
$206.67
So our expected monetary
value for choosing an envelope is two hundred and six dollars
and sixty seven cents ($206.67).
Now, what would you do? Would you take the cash,
expected monetary value of ninety three dollars,
or take an envelope, expected monetary value of two hundred sixty dollars and sixty
seven cents ($206.67)? Now you're never going to get that
amount of money
because you're gonna get five hundred, a hundred, or twenty dollars,
but because the expected monetary value is higher you should probably take the
envelope.
Now, this is a game of chance here, etc, I am an extremely
low risk person, so I still would probably take the cash, cash in hand is better,
because with my luck, I'm going to end up with the twenty dollars.
But you need to decide this for yourself!
So this is the end of our decision sheet problem
in Supplement C
dealing with Slides 7 and 8.