Expected Value

Expected Value

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
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

100 thoughts on “Expected Value”

  1. I was pretty sure I understood the gist of this while working on a problem but was double checking my work. Stumbled here and you clarified it. Thanks!

  2. Never really paid attention in stat class….. exam in about a week….No problem I got patrickJMT to help me 😀

  3. Well, I failed stats last year and I am doing everything – even YouTube – do pass this year. Thank goodness I clicked on your videos. Our professor should watch your videos to see how it should be explained.

  4. Isn't the lower half of the table supposed to be a density function? In which case all the probabilities would have to add up to 1? It seems more logical to leave out the 100% chance of losing a dollar and just subtract it at the end. So the game has an E(X) of 1.875, but it costs a dollar to play so the net gain is 0.875.

  5. Thanks for the help – these vids really do help. Question for you, how much WOULD you pay to play this game…assuming a risk:reward ratio of 1:3?

  6. And so the day before the midterm, not only did I learn that there was a God, but that he posted youtube tutorials on statistics.

  7. At first it seems like getting $0.875 for spending a $1 is a bad choice. Is the right way to interpret this Expected Value problem as:

    "I'm spending $1 to receive an average of $1.875 per spin, thus netting $0.875?"

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  12. How do you do this if you aim to win the 10 dollars in 4 tries? Then what will your expected outcome be? I can do this basic stuff, but I just recently got hit with an intense version about shooting darts and I can't figure it out for the life of me.

  13. You're my favorite math tutor on Youtube. I've passed my math classes thanks to you. I just wish you had more stats videos!

    Thank you!!

  14. First let me say I wish you were my math teacher! Second thanks for all your videos they are getting me through my advanced math in university! Can you help me with this:

    your friend bets you $20 that he can pull 2 spades in a row from a deck of 52 cards (which contains 13 spades). What is your expected value from this bet?
    a. 17.65
    b. 17.76
    c 18.24
    d 18.57

  15. I find it more intuitive to subtract the dollar spent to play, into the winnings yielding …

    outcome    :   -1  :      0     :    1     :     9
    probability :  0.5  :  0.125  :  0.25  :  0.125

  16. Dear Patrick, thank you for this video!
    I've one question . suppose you are required to pay 1$ each time you plan ( not paying one time 1$)  how you describe this on that Outcome&Probability table?

  17. i have been doing some search on this topic for a while, and this is probably the best explanation i have seen. good job

  18. So u have to explain to us that in this example the arrow will always fall on a value and not on a line but u decide not to explain what Xn and Pn mean?

  19. HELP!
    If you know that 35% of university students require vision correction (glasses or contact lenses) and assume that enrolment in classes is random,
    (a) How many students would you expect to require vision correction in a class of 40?
    (b) What distribution would the number of students in a class of 40 that require vision correction follow? Why?
    (c) What is the probability that less than 10 students in a class of 40 require vision correction?

  20. A winning straight ticket earns $5,000 and the winning box ticket wins $2,000 for Cash-3, what is the expected value for each type of bet if each ticket costs $1 and you either win or you lose?

    Anyway you could help me out with this?

  21. ok so does this mean you win $1.875 per game then? otherwise you are loosing money because it costs a dollar to play?

  22. But you said you don't win back your original dollar? So if you played 100 times and won $87.50, you'd have lost money?

  23. Why Expected Value can be used in statistics, people choice is not a possibility, by why when doing chi square test we involved Expected Value in it..?

  24. a bag contains 4 red coloured beads and 6 green coloured beads. Two beads are drawn, one after the other, without replacement. Suppose X is a random variable which represent the number of red beads drawn, find E(X).

    anyone can help to solve?

  25. Very nice video. But I don't understand why -1 has to be considered. Isn't the notion of losing $1 captured by the sector having $0?

  26. The pie example through me off, I thought since 0 was an option twice it would be 2/5 which would give you a probability of .4, and everything else a probability of .2

  27. Dear Patrick- Thanks for the video as always helpful. However, I believe that E(X) is independent of the number of times the game is played so whether you play once or 100 or 1000 times the E(X) will remain 0.875.

  28. I really wish you had some expected value vids for advanced graduate level math like stochastic processes…those are such a pain :/

  29. If you win a dollar, you just get your dollar back, so that's (0*.125) or one-eighth of the spinner. Zero times anything is zero so you can leave that off if you want.The chance that you would lose your dollar is fifty percent, or expressed as a probability is .50 since the zero takes up half the spinner (-1*.50). The 10 dollar space pays off only 9 dollars since you don't get your dollar back, and that is one-eighth of the spinner, or .125 as a probability (9 .125) The 2 dollar pays only 1 dollar since you don't get your original dollar back or (1.25)

    (0 .125)+(-1.50)+(9*.125)+(1*.25)=.875

  30. Hi patric.

    I have truble understanding the analysis at the end. If the player playes 100 times and the E(X) = 0.875, the player is expected to win $87.5. At the same time the the player must spend $100 to play 100 games, wouldnt that mean that the player is loosing $100 – $87.5 = $12.5? Or does it consider the 1 dollar spendt per game so the gain is actually $187.5?

    Great videos btw.

  31. Hi, playing 100 times may win $87.50 on average but the guy have to pay $100 to play 100 times. Then how will this be a profit. Please reply

  32. there is 5 spaces yet you said 1/4 prob + another 1/4. This should be 2/5 should it not?
    Onto another video..

  33. so he'd be in loss,on average …since $0.875 is how much he'd win for every $1 spent, that a 12 cents loss on average.

  34. Can I have a question.please do answer me

    Suppose that you are playing a game with single die assumed fair. In this game, you win Rs 20 if a 3
    turns up. Rs 40 if a 5 turns up, lose Rs 30 if a 6 turns up. You neither win nor lose if any other faces
    turn up. What is the expected sum of money you win?

  35. I don't understand why you should play the game. It costs you 1 ticket every try and on average your expected return is 0.8$. Doesnt that mean ure losing 0.2$ every game?

  36. Hi Patrick. The table you created is not a discrete probability distribution table, correct? If not, what is it called? For the sum of the p(x) ≠ 1. thx

  37. I don't see why this game is profitable for the ones playing ? On average you win 0.875 cents per game, but to play the game you need to give 1dollar, so on average you lose money : 0.875 – 1= -0.125cents

  38. so if I played 100 times spending $100, would ! win $87.50 back but lose $13.50, or would i win back the $100 plus $87.50?

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