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
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