Expected Value - YouTube

Okay in this video. I'm going to talk [about] finding the expected value [of] a data set that has finitely many outcomes

So my outcomes here labeled x sub 1 x sub 2 up to x sub n

And these will occur with probability p sub 1 piece of 2 up to p sub N

Respectively, and it says the expected value of your data set that's what the x represents [basically] it's it's sort of out

you can think about it as being a weighted average or a

Long-Run average and all it all you have to do to compute it is

You take your outcome multiply it by its respective probability of occurence add all of those together and hey, that's your expected value

So I'm going to do one here in conjunction [with] a game ok

so suppose your friend comes up to you and offers to play a game and

So maybe you'll play maybe you won't and maybe your friends not so good at statistics

and he doesn't really he or she doesn't really know whether

you know whether they [should] be playing the game or not, but you'll be clever enough [to] figure it out, so

supposed to play the game it only costs one dollar and

Forgive my bad artistry so suppose you have like a little [a] little spinner, okay?

So that's what the circle is and I've tried to divide it into four equal regions. So again forgive my artistry

[so] you're going to spin the little blue spinner and whatever you know the [arrow] [is] pointing at whatever region?

It's in you'll get that amount of money

and for

Simplicity's sake let's just assume that this it will never fall on a line

You can always decide it falls into one region or the other region

Okay, so the outcomes here are you can win $0 $2 $1 $0 or $10?

Okay, and if you win the game you [know] you don't get your initial $1 back

You just get so you pay a dollar and then your friend will pay you whatever amount is shown

Okay, so a couple things here

we need to list all of our outcomes and

the probability associated with each of those outcomes

Okay, so let's see here. It looks like you can win

[$0] if it falls in the top left corner and also in the bottom right to me

It looks like you know just based on the area of the circle

The top left portion would be [one-fourth] of the circle

[the] bottom right portion would be another 1/4 of the circle so to me. It looks like you could win

$0 with a probability of 1/4 plus 1/4 or

1/2 so there's a 50% chance. It's going to fall in one of those two regions

So you'll win $0 I can win a single dollar

so if this whole entire region represents 1/4 of the circle well if I divide that by 2

each Little region

Will have area 1/8 of the circle?

Okay, so it says the probability of me falling in the region where I would win one dollar would be 1/8

likewise the probability of me winning [$10] would have

probability 1/8 and I think the only other

Possibility would be to win $2 and again that takes up 1/4 of the circle

So the probability that I would win $2 is 1/4

Okay, so notice I [left] a little space [here] at the beginning

One of the outcomes for sure is that [you're] going to lose [$1]

with a probability of [one]

okay, and basically what this

[represents] this just factors into the fact that well it costs

$1 to play the game

Okay, so let's see what what the expected value of this game is

Okay, so all it says is again if we call our data set x it says we're going to lose [$1] with 100%

Certainty we add to that okay. We'll take zero times one half so again. I'm just multiplying

The Outcomes by their probabilities plus one times [1/8]

plus 2 times 1/4 plus 10 times [oops]

Plus 10 times 1/8

Okay, so now all that's left to do is to basically compute the value, so I'll get negative [one] [zero] times 1/2 is zero

Plus 1/8

It looks like I'll get plus 2 over 4 and then

[Plus] 10 over 8

So it looks like I'm going to get common denominators here, so it looks like 8 is what we'll use

So I'll make that negative 8 over 8 and if I multiply top and bottom

Of my other fraction I'll get 4 eighths, so now all I have to do is add these all up

It says you get negative 8 plus 1 which is negative 7 negative 7 plus 4 [is] negative 3 negative

3 plus 10 is

7 so we get the [value] 7 8 and

You know the important thing here is

What does this mean?

Okay, so seven eighths is the number

Point eight seven five so what it says is it says you can expect

to win

On average this [is] the important part

You can expect to win on average

eighty seven and a half cents per game

Okay, so again. This is why I say on average you know notice

The only thing that can happen is you definitely lose your dollar?

[and] then sort of the positive outcomes as you don't either you win zero dollars one dollar two dollars or ten dollars

Okay, it's not possible to win point eight seven five dollars per game, but again

It's a long-run average and what it means is on the whole you're going to win money

so if your friend offered to play this game, you would say

Absolutely, I would play this game

Suppose you played 100 times

so if you play 100 times

You could expect to win

You could expect to win

0.875

times

100 so that would simply move the decimal place twice or give you 87 50, so if you can expect to win

[0.875] dollars each time you play you could expect to win Roughly eighty seven and a half dollars if your friend

Was crazy enough to play with you for that long, [so] this is the basic notion of expected value

So it [represents] an average somehow weighted average so all right. I hope this [example] makes some sense

So you know don't don't offer to play this game with somebody where you're the one charging a [dollar]?

for sure, so

Again, I think it's a nice [little] illustration

I kind of use these game examples just to remind myself of

What expected value is I think it gives you kind of a good little intuitive idea of what's going on so all right again

I hope this helps if you have any questions, or comments, please feel free to post them as always