4.3 Game Expected Value (Example 2 & 3)

All right. So the next piece is,

in general, if the expected value

of a game is negative,

you can expect to lose money

over the long run.

We're talking about a game,
that's why we're talking

about winning money
or losing money.

If the expected value
is positive...,

you can expect to win...

money over the long run.

So to go back
to the previous example,

you know,
our expected value was $1.40,

so we're expected
to win that on average.

The difference is we just haven't
taken into effect

how much we actually paid
to play the game.

And so that's why
that became negative,

that $0.60.

Now if the expected value is zero,

it is considered a fair game...

since neither the player
or owner are expected

to come out ahead
over the long run.

So casinos definitely
do not play a fair game.

[laughs]

All right.
So let's do Example 2.

Let's talk about another game.

A roulette wheel
has 18 black numbers,

18 red numbers,

and then the numbers 0

and 00 in the green.

In roulette, betting on a red
is a 1 to 1 bet.

And so what that means
is if you put $5 down,

you're going to win $5,
or lose $5.

Okay, so whatever you put down,
you gain or lose.

That is if you win.

I just said that [laughs].

Suppose you make a $10 bet on red.

What are your
expected net winnings?

So what we're going to do here is,

we are going to create
a probability model,

or we can call it
a probability distribution.

So I'm going to make a list.

I have my outcomes.

Forgive that writing,
it's atrocious.

Outcomes

and then we have

our net winnings.

And then we have the probability
of it happening.

Okay, so our outcomes here

are we either get a red...

or we don't get a red.

So, not red.

That's the whole process
of how this fun game works.

[laughs]

Now if you land on a red,

you will be up... $10.

And that's why I said net winnings,
okay.

So we're not taking into effect
that we put down $10.

We're going to win that $10.

Then if you don't win on our red,
we're out $10.

That whole 1 to 1 bet thing.

Now if we have 18 black,

18 red,

and then two of these guys,

total...,

we have 38 possibilities.

Well, there are 18 ways
we can land on red.

So that's 18 out of 38.

And not red is--
well, it is the 18 red

and then these two greens,
because they're also not red.

And so that's why
that would be 20 over 38.

And one really nice thing is,
to make sure that you got it right,

is to add these up

and make sure it comes to 1.

And it sure does.

All right,
so let's do the expected value.

Well, I'm going to do expected value.

It would be plus $10,

and there's 18 out of 38 chances

of getting that.

PLus a negative $10,

and there's a 20 out of 38 chance

of getting that.

And when we do that in our calculator--

And again, I'm just relying
on my handheld calculator

just to not switch screens
a million times.

I'm going to go ahead
minus ten times the 20 over 38.

And this is money,
so I'm going to round it to the tenths.

Make sure you just pay attention
to how, um, my open math wants it.

And it looks like about $0.53.

And so what that's telling me--
because it's negative,

okay-- we can expect...

to lose--

And I don't have to put the negative

since I use the word use--
$0.53 per game.

And again, to be really clear,
over the long run.

This does not mean you play

and you're going to lose
$0.53 your first play.

So if you sit there all night
and you keep playing,

and you knew you should
have walked away,

you're probably going
to end up losing.

So what I'm going to ask you now to do

is to see if it's sitting well with you,

because example three
is pretty straightforward.

I pause this video

and see if you can do example three,

um, and then start playing it again
and see if you got the right answer.

All right, so a bag
contains 3 gold marbles,

6 silver marbles,
and 28 black marbles.

Someone offers to play this game.

You randomly select a marble
from a bag.

If it is gold, you win $3.

If it's silver, you win $2.

If it's black, you lose a dollar.

What is the expected value
of this game?

So I need to come up
with a probability distribution.

So just like we just did,
here my outcomes,

net winnings,

probability.

And I probably didn't
have to make this so big

so I could actually write something.

All right, well total,
3 plus 6 plus 28 is 37.

So I have options of pulling

a gold, or a silver, or black.

Well if I pull a gold
it looks like I win $3.

So I'm going to put a plus there.

And my probability
would be there's only three golds

but there's 37 total in the back.

Silver I win $2

and that is 6 out of 37.

And then black I lose the dollar,

and that is 28 out of 37.

So of course losing

has a bigger chance of happening.

And again, when you're doing
this on your own,

just to make sure it's correct,

add those all up.

Make sure you get 37 out of 37-- and I do.

And so my expected net winnings--
or expected value,

however you want to say it.

I like net winnings
because then I know it's exactly talking

about what you're winning.

It's going to be
$3 times that probability

plus $2 times that probability
plus a negative dollar--

Or you could just write minus--

of that probability.

I'm putting approximately because
I'm going to have to do some rounding.

So again, I'm going to my calculator.

Whatever one you like to use because you

always are allowed to use a calculator
in this class.

And I'm going to do
plus 2 times 6 out of 37

minus one times 28 out of 37.

And again, because it's money,
I'm going to round to the hundreds.

It's negative, and it's a money, $0.19.

So again, nice little sentence.

On average,

you can plan

to lose

$0.19 per game.

Again, being really specific
over the long run.

Okay, so that's really good to know.

Not that I ever want
to encourage gambling,

but hopefully this
has encouraged you maybe not to

because you were seeing
we just constantly are losing money.

But to also just, like, walk away.

All right.

The next, um, couple examples
I'm going to put

in the next video
because it's going

to be talking
about insurance companies.

All right, see you next.