### Expected Value in Gambling

Okay in this article, 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.