We've talked about using probability to 
determine what to believe. 
Such as, whether to believe that a 
certain form of contraception will work 
or will probably work. Or whether to 
believe that you have cancer when a 
doctor has done a test and you got a 
positive result which can be pretty 
scary. 
But you still have to figure out whether 
to believe that you really do have 
cancer. 
Now, probability can be very useful in 
the formation of beliefs. 
But it's also extremely useful in making 
decisions. 
So, in this lecture, we want to talk 
about the use of probability to make 
decisions or choices. 
There are three kinds of decisions or 
choices that we need to discuss. 
First of all, decisions with certainty. 
Second, decisions with risks. 
And third, decisions with ignorance. 
Or, as those are sometimes called, 
decisions with uncertainty. 
And we'll talk about the first two and 
then I'll just say a few words about the 
third. 
First, let's look at decisions with 
certainty. 
A decision with certainty is one where 
you know exactly what's going to happen, 
you know what the effects are going to be 
and you know it definitely. 
Its not that its probably going to 
happen, 
you know what's going to happen. 
You know the good effects and the bad 
effects. 
So there's no uncertainty, everything is 
certain. 
For example, suppose you go to a 
restaurant, an Italian restaurant, with a 
friend of yours. 
And, you have to decide whether to order 
pasta or pizza. 
Some really nice eggplant Parmesan on top 
of pasta that you like at this 
restaurant. 
They also have really good mushroom 
pizza. 
So you have to think about that, but you 
know that if you order the eggplant 
Parmesan pasta, then they're going to 
bring it to you. 
And you know that if you order the 
mushroom pizza, they'll make one for you, 
and you've got enough money for both of 
them so there's no uncertainty. 
You know what you're going to get. 
Still might be a difficult choice 
because, well, you might not be sure 
whether you feel more like the eggplant 
Parmesan pasta tonight or the mushroom 
pizza. 
And, they're going to be also potentially 
a conflict of values. 
Maybe the eggplant Parmesan pasta costs 
more than the pizza. 
In which case you have to decide, 
well, even though I like it more, do I 
want to spend that money right now? 
You got a conflict between what you feel 
like and how much it's going to cost. 
Or, there might be a conflict with your 
friend. 
Your friend wants to buy the mushroom 
pizza and split it with you because he 
doesn't want an entire pizza himself. 
But you really feel like the eggplant 
Parmesan. 
So now, you've got a conflict between 
your loyalty to your friend. 
You want to support your friend and help 
your friend get what he wants. 
And yet, you feel like the eggplant 
Parmesan yourself. 
So, these decisions can be very difficult 
in some cases but there's no uncertainty. 
You still know that if you order the 
eggplant Parmesan you're going to get 
eggplant Parmesan, 
or if you order the pizza you're going to 
get pizza. 
Right? 
There's not uncertainty at all. 
No, of course there's uncertainty. 
You might order the pizza and it comes 
burnt. 
Yuck. 
Oh, I hate burnt pizza. 
And you might order the eggplant Parmesan 
and it turns out that they really messed 
up the sauce, and it's got way too much 
salt in it tonight. 
Maybe they have a new chef in the kitchen 
who's going to mess up one but not the 
other. 
It's not really certain that when you 
order it, you're going to get what you 
wanted or that you're going to like it. 
Now, this is just a fact about life. 
There's nothing really certain in this 
life. 
You might think that you know exactly 
what's going to happen. 
But there's always at least some chance 
that it's not going to turn out the way 
you thought. 
And so, what do you do about that? 
Well, some people just stipulate. 
If I order the pizza I'll get the pizza. 
And it'll be good pizza. 
Well, you can of course stipulate that. 
But then, it just makes the whole 
situation unrealistic. 
Cause in the real life, you're never 
going to know for certain. 
So another possibility is to say, we're 
just going to ignore these slight chances 
that the pizza's going to come back 
burned. 
I've been at this restaurant a hundred 
times, before they've always done it just 
right. 
And so, what are the odds they're going 
to burn it this time? 
Pretty slim. 
99% chance they're going to do it the 
right way. 
Well, you might want to say, I'm just 
going to ignore the 1% chance of getting 
burnt because it's not going to make any 
difference. 
I still want the pizza or I still want 
the eggplant, 
whichever. 
The other thing you might do is say, 
look, nothing's certain, but the 
probabilities are balanced. 
They go on both sides. 
After all, there's a chance of burning 
the pizza, 
there's also a chance of burning the 
eggplant. 
So, in some cases it might make perfectly 
good sense to ignore these uncertainties 
because they're not going to effect, your 
decision making process. 
Still, it's important to realize that in 
life, throughout life, 
in all areas of life, there's always 
going to be some uncertainty. 
And that means, there's going to be a 
probability that if you look at it 
seriously, 
it's going to play into the decision 
making process. 
So, 
the kinds of cases that we want to focus 
on are the cases of risk where there is a 
probability of failure and a probability 
of success for the different plans that 
you have to choose among. 
All realistic examples then are actually 
decisions under a risk, especially the 
most important ones. 
So, what is a decision under risk? 
It's simply a case where you're not 
absolutely certain what's going to 
happen. 
It's not definitely this effect will 
occur, that effect will occur. 
You'll get your pizza, 
it'll taste good. 
You'll get your pasta, 
it'll taste good. 
Instead, there's a probability of each 
outcome. 
There's a probability that the pizza will 
get to the table. 
There's a probability that the pizza will 
taste good. 
There's a probability that the pizza will 
never make it to the table but get 
dropped by the waiter, or that the pizza 
will taste horrible because it was burnt. 
Then, you have a decision under risk. 
And how do we make these decisions under 
risk? 
Well, there are a lot of different 
theories,. But one, pretty common one, is 
to apply expected value theory. 
Now, we're going to see that there are 
lots of different values, and it's hard 
to see how to handle them all, but let's 
start with the really simple case. 
Let's assume for now that all that 
matters, is money. 
One common way, perhaps the most common 
way, of figuring this out is to calculate 
the expected monetary value or the 
expected financial value. 
And that calculation is pretty simple. 
It's the probability of winning times the 
net gain if you do win, 
minus the probability of losing times the 
net loss if you do lose. 
And here we're talking about gains and 
losses in terms of money. 
But why do we say net loss and net gain? 
Imagine, you buy a lottery ticket. 
And you win $20 million and they sent you 
a check for $20 million Well, how much 
did you win? 
Sounds like you won $20 million but not 
really. 
You never got that dollar back that you 
paid for the ticket. 
So really, you only won $19,999,999. 
You know, that's pretty trivial if you 
just won the lottery. 
No big deal. 
But now, imagine a different 
circumstance. 
When you're playing poker and you bet 
$10. 
And the opponent raises it to twenty. 
And you're bluffing so you want to bet 
big. 
So you bet $200. 
Put that all in the pot, too. 
And your opponent backs down and you pull 
in the entire pot. 
How much has it got in it? $230. 
Well, did you win $230? 
No, because you put in 210. 
You only won twenty. 
The net winnings are how much you pull in 
minus how much you put in before you 
pulled in your winnings. 
So it's 230 minus the 210 you put in, 
you only gained twenty. 
A lot of people forget that in poker and 
they end up thinking that they win a lot 
by bluffing when actually they didn't win 
as much as they thought they did. 
In any case, we have to look at the net 
gain and the net loss when applying this 
formula for expected financial value. 
Let's do a few examples of calculating 
expected financial values, and let's do 
them with a deck of cards because we're 
familiar with that. 
There are thirteen different cards and 
four different suits, spades, hearts, 
diamonds and clubs. 
So we have to remember that to calculate 
the probabilities. 
Let's imagine that you can make this bet. 
If you bet a dollar, you'll win $5 net if 
you pick a spade. 
Now, notice that it's net. So that means 
either you put a dollar in, and then you 
take $6 out, and your net gain is five. 
Or you say, if it's not a spade I'll give 
you a dollar, and if it is a spade you 
give me $5. 
Either way, if you pick a spade, you're 
going to be $5 up. 
What's the expected financial value of 
this bet? 
Well, we know that the odds of picking a 
spade is one in four. 
That's the odds of winning because if you 
pick a spade you'll win. 
And the net gain is $5. 
So, to calculate the expected financial 
value, we need to multiply this 
one-quarter, the odds of picking a spade, 
times five, the number of dollars that 
you win, net if you win. 
And then, we have to subtract the odds of 
loosing three quarters, because if you 
pick a heart, a diamond, or a club, then 
you loose, and what do you loose? 
You loose $1. 
So, one quarter times five is five 
quarters, three quarter times one is 
three quarters, subtract three quarters 
from five quarters, you get two quarters 
and that of course is a half. 
So the expected financial value of this 
bet is one-half. 
Now, compare that bet to another bet. 
In this bet, if you bet a dollar, you're 
going to win ten. 
That sounds pretty good. 
Instead of winning five, you're going to 
win ten. 
But, you only win $10 net if you pick an 
ace. 
And there are thirteen cards, so we have 
to redo the calculations. 
And the probability of picking an ace is 
one in thirteen. 
The amount you win is ten. 
That's your net winnings, remember. 
You have to subtract the probability of 
losing, twelve in thirteen because you're 
going to lose if it's any card other than 
an ace. 
And the amount you'll lose is $1 net. 
10 * 13 = 13/10. 
Subtract 12/13 * 1, which is 12/13, and 
you get -2/13. 
Notice that's -2/13. 
That means the expected financial value 
of that bet is negative. 
You ought to expect to lose money over 
the long run if you play that kind of bet 
a lot. 
Third example. 
This time you can bet a dollar and win 
$51 net if you pick an ace of spades. 
But you can't pick any old ace, or any 
old spades. 
It's got to be the ace of spades. 
So now, what's the financial value of 
that bet? 
The expected financial value is 
probability of winning one over 52, times 
the amount you win net, which is 51, 
minus the probability of losing 51 over 
52, times the amount you lose net, if you 
lose. 
51 * 1/52 = 51/52. 
51/52 * 1 is 51/52. 
Subtract them and you get zero. 
So in this bet, the expected financial 
value of the bet is zero. 
Okay? 
Now, we can compare these two bets by 
asking what bet should you make. 
Well, the first bet was favorable. 
A bet's favorable if its expected value 
is greater than zero. 
And that means that it's in your interest 
to bet, at least if your only interest is 
money. 
A bet's unfavorable if its expected value 
is less than zero. 
And a bet is neutral or fair if its 
expected value is zero. 
And if you have a bet that's favorable, a 
bet that's unfavorable, and a bet that's 
neutral, it shouldn't be surprising that 
the one you should pick is the one that's 
favorable where the expected value is 
greater than zero, if you're going to bet 
at all. 
And that's the next question that we have 
to turn to.