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1. ## An introduction to expectations and theoretical hold

In this post I talked about implied win percentages. Today I’m going talk about how to use those percentages to determine how this relates to a book’s expected profit on any given bet. This is known as theoretical hold.

The notion of expectation is central to probability and statistics. Essentially an expectation is just an average with an extra syllable. If you flip a coin 10 times then you expect it will land on heads 5 times and you expect it will land on tails 5 times. In reality of course the coin’s not always going to land on heads exactly 5 times out of 10 (in fact it will only do so about 24.6% of the time), but if you repeat the experiment (flipping a coin ten times) many, many times over then on average it will land on heads 5 times each trial.

The same thought process is also applicable to sports. If the Yankees can be expected to win a particular game 60% of the time, then this would mean that if the exact same game were repeated under the exact same conditions across many, many parallel universes, we would expect the Yankees to win 60% of those encounters.

So let’s say you bet \$1 straight up that the Yankees are going to win that game. Now that’s quite obviously a good bet. But just how “good” is it? Well that’s where expectations come in. If you made the same bet in each of those parallel universes you’d win \$1 60% of the time, and lose \$1 40% of the time. Now let’s say that there are actually 1,000,000 of these such universes. Exactly how much money would you make? Well, in 600,000 of those universes you’d make \$1 for a total of \$600,000 dollars, and in the remaining 400,000 of those universes you’d lose \$1 in each game for a total of \$400,000 dollars. So you'd receive \$600,000 and would pay out \$400,000 meaning that your total profit would be \$200,000. Winning \$200,000 across 1,000,000 means on average you would have won \$200,000 / 1,000,000 games = \$.20 per game.

Now of course 1,000,000 is just a made up number in this context. There aren’t really 999,999 other universes where we could make such a bet. This bet can only be made once. But that doesn’t actually matter in the world of statistics. Whether you can make this bet only one time or you can make it multiple times the expectation per game is precisely the same, namely 20%.

So in general the way you calculate the expected profit (or loss) of a bet is with the following formula:
Code:
```E(Unit Profit)
= probability of win * amount won – probability of loss * amount lost1```
Using our example from above of a 60% probability of a win for a straight up bet, E(P) = 60% * 1 unit – 40% * 1 unit = .2 units. Because the player was risking 1 unit, his % expected profit is just 0.2 units / 1 unit risked = 20%. So the formula for Expected % Profit is:
Code:
`E(P) = probability of win * amount won/amount lost – probability of loss`
You’ll recall that when we talked about implied winning percentages we said that the implied probability of a line set are those probabilities that would equate the player’s expected losses from betting on either side of the event. This expected loss figure is known as theoretical hold (it’s actually the negative of theoretical hold) and has a special significance in sports betting. It corresponds to the profit a book would expect were a player to bet (either side) of an event with all else being equal. So for example in the case of a -110 line set, we known that the implied single line probability is 110/210 ≈ 52.38%, and hence the line set probability is just 52.38%/(52.38% + 52.38%) = 50%. Using the formula from above we see that the expected player loss = 50% * 100 units/110 units – 50% ≈ -4.55%. In other words the theoretical hold of a line set offered at -110/-110 is 4.55%.

A theoretical hold of 4.55% means that the book’s expectation on a bet placed on either side is 4.55%. Just like in the coin flip example, this doesn’t mean that a book always expects to make 4.55%, just that that’s what the book expects to make on average.2 The methodology given above for calculating theoretical hold above is certainly serviceable: Calculate the individual zero-vig implied probabilities, use those to calculate the line set probabilities, and then finally plug the line set probabilities and the original lines into the expected value formula to come up with an answer. Nevertheless, it's also a little arithmetically involved. But there is actually a slightly easier way.

Recall that overround is just the sum of the zero-vig probabilities. (So in the case of -110/-110 the overround would just be ≈ 52.38%+52.38% = 104.76%.) Sparing you the algebra the formula for theoretical hold is given by:
Code:
`Theoretical Hold = 1 – 1 / overround`
.

So you’ve probably noticed that as lines increase in magnitude nominal spreads (dog line + fave line) also tend to increase. A book that will offer lines at -105/-105 might also offer lines at -210/+190 or -1000/+800. But now we can actually figure out exactly how much we expect to pay in vig for each line set. So:
Code:
```-105/-105:
overround = 105/205+105/205 ≈ 102.44%
theoretical hold ≈ 1 – 1/102.44% ≈ 2.38%

-210/+190:
overround =  210/310 + 100/290 ≈ 102.22%
theoretical hold ≈ 1 – 1/102.22% ≈ 2.18%

-1000/+800:
overround =  1000/1100 + 100/900 ≈ 102.02%
theoretical hold ≈ 1 – 1 /102.02% ≈ 1.98%```
So this means that the 10c wide spread at -105/-105 is more expensive (by about 9.4%) than the 20c wide spread at -210/+190 which is in turn more expensive (by about 9.9%) than the 200c wide spread at -1000/+800. Hence, contrary to at least quasi-popular opinions, larger nominal spreads don't necessarily imply greater profitability for the book.

1. It works the exact same way in the case of a multi-way bet with multiple possible winning and/or losing outcomes. Expected profit is just given by p(win type 1) * amount won(outcome 1) + p(win type 2) * amount won(outcome 2) + p(win type i) * amount won(outcome i) + … - p(loss type i+1) * amount lost(outcome i+1) - p(loss type i+2) * amount lost(outcome i+2) - … .
2. Although a book would expect to make 4.55% in the case of balanced action. Balanced action does NOT necessarily mean an equal amount bet on each side (it would only mean that were both sides of the bet offered at the same line, such as, say, -110/-110 or -105/-105) but rather that the amounts bet on each side are in proportion to the implied line set win percentage. For example, given a line set of -210/+190, balanced action would imply about 66.27% (the implied favorite win percentage) of bet value coming in on the favorite.
Points Awarded:
 Dave Head gave ganchrow 5 SBR Point(s) for this post.

SBR Founder Join Date: 8/28/2005

2. WTF???

Ganchrow ... naked pictures of Christina Ricci will get 1000 posts ... but this is hilarious.

Can you imagine, say, slash trying to understand this? Its got no pictures for a start so thats 98% of the forum passing by.

That said, i swear i will read it (Im tired now ... World Cup amnd all that) and get back to you with my impressions as this stuff is important and I should get my head round it.

SBR Founder Join Date: 9/14/2005

3. Originally Posted by natrass
Ganchrow ... naked pictures of Christina Ricci will get 1000 posts ... but this is hilarious.
Wish I could help out with that one but unfortuantely she didn't let me bring my camera.

SBR Founder Join Date: 8/28/2005

4. I'm just hoping that in one of the parallel universes you mention I'm a helluver lot smarter there and can understand all the math!

SBR Founder Join Date: 12/23/2005

5. If it's not undestandable then it's the fault of the author. Mea culpa.

If you have any questions feel free to ask away.

SBR Founder Join Date: 8/28/2005

6. Originally Posted by ganchrow
If it's not undestandable then it's the fault of the author. Mea culpa.

If you have any questions feel free to ask away.
No it's not, I just didn't get the proper edumaction and probably bunked-off (played hooky) one too many times me thinks!

SBR Founder Join Date: 12/23/2005

7. it's amazing how that brain of yours works, Ganchrow...

SBR Founder Join Date: 8/18/2005

8. Originally Posted by natrass
WTF???
Can you imagine, say, slash trying to understand this? Its got no pictures for a start so thats 98% of the forum passing by.
Great post natass - without boasting of my own intelligence, I would say that I am in top 1% on this forum and top 5% in general.

I would imagine that the opposite would be a fair guess of your abilities.

SBR Founder Join Date: 8/10/2005

9. Can you show us some pictures?

SBR Founder Join Date: 12/16/2005

10. Originally Posted by slash
Great post natass - without boasting of my own intelligence, I would say that I am in top 1% on this forum and top 5% in general.

I would imagine that the opposite would be a fair guess of your abilities.
Fair enough.

No, but wait .... how can you know what the average intelligence of the people on this forum is?

You can't. So, to make such a comparitive claim is, well, unintelligent surely?

SBR Founder Join Date: 9/14/2005

11. Bump

13. In a parallel universe could you have different results due to probability?

14. I miss Ganch.

Some say he ended up broke, living on some beach.

SBR Founder Join Date: 12/14/2005

15. Originally Posted by Dark Horse
I miss Ganch. Some say he ended up broke, living on some beach.
I wouldn't mind being this sort of broke.

Regardless, I miss him as well. He was always extremely (perhaps excessively?) helpful to a variety of asinine questions. Not to mention his math was uber studly.

1. Although a book would expect to make 4.55% in the case of balanced action. Balanced action does NOT necessarily mean an equal amount bet on each side (it would only mean that were both sides of the bet offered at the same line, such as, say, -110/-110 or -105/-105) but rather that the amounts bet on each side are in proportion to the implied line set win percentage. For example, given a line set of -210/+190, balanced action would imply about 66.27% (the implied favorite win percentage) of bet value coming in on the favorite.

THIS IS PRECISELY WHY ONE NEEDS TO ALWAYS BE AWARE OF THE LINE IF THEY ARE AT THE VARIOUS SITES THAT PROVIDE THE PERCENTAGES BET ON A PARTIULAR TEAM.

SBR Founder Join Date: 8/11/2005

16. bump

18. Originally Posted by Fishhead
1. Although a book would expect to make 4.55% in the case of balanced action. Balanced action does NOT necessarily mean an equal amount bet on each side (it would only mean that were both sides of the bet offered at the same line, such as, say, -110/-110 or -105/-105) but rather that the amounts bet on each side are in proportion to the implied line set win percentage. For example, given a line set of -210/+190, balanced action would imply about 66.27% (the implied favorite win percentage) of bet value coming in on the favorite.
THIS IS PRECISELY WHY ONE NEEDS TO ALWAYS BE AWARE OF THE LINE IF THEY ARE AT THE VARIOUS SITES THAT PROVIDE THE PERCENTAGES BET ON A PARTIULAR TEAM.
I've been trying to understand what benefit the books gain by sharing their market % info with places like Sports insights, this nails it. ty

19. Originally Posted by Aurelius

I've been trying to understand what benefit the books gain by sharing their market % info with places like Sports insights, this nails it. ty
It is beneficial if used correctly.
175 pts

3-QUESTION
SBR TRIVIA WINNER 05/24/2018

20. darn i thought ganchrow was back

SBR Founder Join Date: 10/23/2005

21. sounds good to me

22. Wow, I really miss these posts. Does anyone know where Ganch went?

23. Originally Posted by Wrecktangle
Wow, I really miss these posts. Does anyone know where Ganch went?

I dont know,but you do look alot like swinging johnson.

24. What does this have to do with picking winners ?

SBR Founder Join Date: 10/30/2005

25. Originally Posted by Sam Odom
What does this have to do with picking winners ?
Did you even read it?

26. Originally Posted by Wrecktangle
Wow, I really miss these posts. Does anyone know where Ganch went?
He posted the other day around 4am.

27. Originally Posted by durito
He posted the other day around 4am.
Where, here at SBR somewhere?

28. Tag

29. Originally Posted by Ganchrow
Now of course 1,000,000 is just a made up number in this context. There aren’t really 999,999 other universes where we could make such a bet. This bet can only be made once. But that doesn’t actually matter in the world of statistics. Whether you can make this bet only one time or you can make it multiple times the expectation per game is precisely the same, namely 20%.

So in general the way you calculate the expected profit (or loss) of a bet is with the following formula:
Code:
```E(Unit Profit)
= probability of win * amount won – probability of loss * amount lost<sup>1</sup>```
Using our example from above of a 60% probability of a win for a straight up bet, E(P) = 60% * 1 unit – 40% * 1 unit = .2 units. Because the player was risking 1 unit, his % expected profit is just 0.2 units / 1 unit risked = 20%. So the formula for Expected % Profit is:
Code:
`E(P) = probability of win * amount won/amount lost – probability of loss`
Can someone please explain to me how he got from the first equation to the second?

30. Originally Posted by keemosabi
Can someone please explain to me how he got from the first equation to the second?
I'm not exceptional at math, but the first equation is how much you can expect to gain/lose per bet (on average) in terms of units (or dollar amount). Hence: "E(Unit Profit)"

The second equation is how much you can expect to gain/lose per bet in terms of percentage. Hence: "E(P)"

Basically they are the same thing just different ways to illustrate it = expected units vs. expected percentage.

31. Originally Posted by thehoorse
I'm not exceptional at math, but the first equation is how much you can expect to gain/lose per bet (on average) in terms of units (or dollar amount). Hence: "E(Unit Profit)"

The second equation is how much you can expect to gain/lose per bet in terms of percentage. Hence: "E(P)"

Basically they are the same thing just different ways to illustrate it = expected units vs. expected percentage.
No I understood the different purpose of each equation, I was just wondering where the second one comes from.

I think if you take the first equation, divide that whole thing by the units wagered, you should get the percentage. I just can't find any way to rearrange the first equation into the second.

32. dude has a mensa mind

33. Not trying to be obtuse but this is not complicated and should be intuitive to any semi serious gambler

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