Expectations and Theoretical hold

[Kramer]

DOOBIE THOUGHT
NFL


Scoring in football tends to take place in football on average only about
8 times a game so, , the final scores do group
around certain numbers, known as key numbers.
Suppose field goals make up (10% X ?) of final margin scores


In the most basic of cases, lets assume there were wagers of $110
to win $100
on the Raiders +3 and then $110 to win $100 on New England -2.5.
The game is balanced but the House would risk being sided if the game
ended with the Pats up 3.

The Oakland wagers would be pushed
but the Patriots wagers would be winners so the House
would lose the $100.
In this case, the House would be in a situation where it could
only lose $100
or collect the vig ($10) if the score fell on any other number.
If the house had stayed at -3 and remained exposed the $100 on Oakland,
a Pats win by 3 would have resulted in a push,
where as any other outcome
would result in either a $100 loss or a $110 win for the House.
Same worst case of a $100 loss, but now the upside is much, much better.
This is why sportsbooks are very reluctant to move off of key numbers.

Math Formula for this ? LoL



The chance of the final score for a game landing on 3 is roughly X %.
Assuming an “even” split between dogs and favorites, the chance of a 3 point
favorite winning by 3 is about X? %.


This would mean that X?% of these games would result in the House
being sided or middled. This would be disastrous for a business where 4.5%
is the profit margin ...

the same risk applies in moving onto key numbers.
If money comes in on the favorites at -2.5 and the dog at +3, the House
would be in the same situation if the score fell 3.

( Remember that when sportsbooks move a pointspread
, the odds don't change, as the spreads only affects whether
your wager is a winner or not. Changing the odds, on the other hand,
doesn't affect whether your wager is a winner or not,
but instead affects the payout.)


If some 1 replies I have no clue
what you're talkin bout
its the doobie..

[Romanowski]

Kramer I moved your post because it was veering off the topic off Market moves with Pinnacle and deserved its own thread.

In this thread we talked about talked about implied win percentages. Now Iets talk about how to use those implied odds from a line to determine how this relates to a books expected profit on any given bet. Also known as theoretical hold.

If the Bears 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, we would expect the Bears to win 60% of those encounters.

So let’s say you bet $100 straight up that the Bears are going to win that game. Now that’s quite obviously a good bet with a 60% expected win.

However how “good” is it?

Well that’s where expectations come in. If you made the same bet in each of many theoretical parallel universes, under the exact same conditions you’d win $100, 60% of the time, and lose, for arguments sake with -110 vig, $110, 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 $100 for a total of $60,000,000 dollars, and in the remaining 400,000 of those universes you’d lose $110 in each game for a total of $44,000,000 dollars. So you'd receive $60,000,000 and would pay out $44,000,000 meaning that your total profit would be $16,000,000. Winning $16,000,000 across 1,000,000 means on average you would have won $200,000 / 1,000,000 games = $.16 per game.

Whether you can make this bet one time or a million times the expectation per game is precisely the same, namely 16%.

So in general the way you calculate the expected profit (or loss) of a bet is with the following formula:

E(Unit Profit)= probability of win * amount won – probability of loss * amount lost

Using our example from above of a 60% probability of a win for a -110 up bet, E(P) = 60% * 1 unit – 44% * 1 unit = 0.16 units.

Because the player was risking 1 unit, his % expected profit is just 0.16 units / 1 unit risked = 16%.

So the formula for Expected % Profit is:

E(P) = probability of win * amount won/amount lost – probability of loss


You’ll recall that in the implied winning percentages thread 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, actually the negative of theoretical hold, and has a special meaning in betting on sporting events.

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 know 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%. 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.

So in short, Calculate the theoretical, individual zero-vig implied odds, 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.

Here's another method with less steps..

Theoretical Hold = 1 – 1 / overround

Where 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%.

You may have 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.

for -105/-105 overround = 105/205+105/205 ≈ 102.44%

so the theoretical hold ≈ 1 – 1/102.44% ≈ 2.38%

for -210/+190 overround = 210/310 + 100/290 ≈ 102.22%

so the theoretical hold ≈ 1 – 1/102.22% ≈ 2.18%

for -1000/+800 overround = 1000/1100 + 100/900 ≈ 102.02%

so the theoretical hold ≈ 1 – 1 /102.02% ≈ 1.98%

So this means that the 10 cent wide spread at -105/-105

is more expensive (by about 9.4%) than the 20 cent wide spread at -210/+190

which is in turn more expensive (by about 9.9%) than the 200 cent wide spread at -1000/+800.

So contrary to what many may have thought a larger nominal spread

doesn't necessarily imply greater profitability for the book

[THE Sports PROFIT]

Kramer...these "Doobie Thoughts" have great possibility ....trademark asap....money...

[Romanowski]

Originally Posted by THE Sports PROFIT

Kramer...these "Doobie Thoughts" have great possibility ....trademark asap....money...

not in this section they dont...focus is key...so perhaps the thoughts/brainstorm while "doobied"

yet written while sober?

kramer, please keep point relative to the thread title

you first asked about the books hold then to what a point is worth in various nfl situations

two totally different concepts..

this recent post goes in a new thread

I have no problem with different ideas in fact its a valuable post but this section is going to be resource and easily indexed, and searchable

how can we find this post, under a thread with a totally different concept?

so once again im moving this to a new thread

[THE Sports PROFIT]

ROME, personally under the influence of Non-doobicus maximus but liquified well....only meaning ...the "whole concept" of doobie thought's could work in a marketable way within a show, column....etc....haven't read a word otherwise....

[Kramer]

Originally Posted by Romanowski

not in this section they dont...focus is key...so perhaps the thoughts/brainstorm while "doobied"

yet written while sober?

kramer, please keep point relative to the thread title

you first asked about the books hold then to what a point is worth in various nfl situations

two totally different concepts..

this recent post goes in a new thread

I have no problem with different ideas in fact its a valuable post but this section is going to be resource and easily indexed, and searchable

how can we find this post, under a thread with a totally different concept?

so once again im moving this to a new thread

No probs Rome ,

When I see a thread called line moves , it could be broken into
various categories

Its kinda VAGUE ,
so not sure where to post some stuff

I have no problem if you move a post I make

For Example , who do you follow
For its line movement?



What general line direction moves make ?

How often are they affected?

What are key numbers in certain sports?

What sports are affected by line movements ?

Why does the book move a line ?


Do you follow or fade?

Stats to back it up

Why or why don’t you follow line movement ?

Line Movement could be one thread
IMO

cheers

maybe I post stuff off
but don't wanna clutter things

but it could be all "relative"
to line movement in general..

[Romanowski]

1. it wsnt a thread called line moves, it was called pinnacle lean

2. of course everything is at least loosely connected, but the last thing we want is a thread called "gambling" with everything in it ( I realize thats an extreme example to get the point the point across)

3. This section is to be highly indexed, and searchable, I cant drive that home enough

which means when youre looking for specific information its easy to find

4. So do clutter...please, its not a problem, make a new thread when its specific info

[Ganchrow]

Originally Posted by Romanowski

Kramer I moved your post because it was veering off the topic off Market moves with Pinnacle and deserved its own thread.

In this thread we talked about talked about implied win percentages. Now Iets talk about how to use those implied odds from a line to determine how this relates to a books expected profit on any given bet. Also known as theoretical hold.

If the Bears 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, we would expect the Bears to win 60% of those encounters.

So let’s say you bet $100 straight up that the Bears are going to win that game. Now that’s quite obviously a good bet with a 60% expected win.

However how “good” is it?

Well that’s where expectations come in. If you made the same bet in each of many theoretical parallel universes, under the exact same conditions you’d win $100, 60% of the time, and lose, for arguments sake with -110 vig, $110, 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 $100 for a total of $60,000,000 dollars, and in the remaining 400,000 of those universes you’d lose $110 in each game for a total of $44,000,000 dollars. So you'd receive $60,000,000 and would pay out $44,000,000 meaning that your total profit would be $16,000,000. Winning $16,000,000 across 1,000,000 means on average you would have won $200,000 / 1,000,000 games = $.16 per game.

Whether you can make this bet one time or a million times the expectation per game is precisely the same, namely 16%.

So in general the way you calculate the expected profit (or loss) of a bet is with the following formula:

E(Unit Profit)= probability of win * amount won – probability of loss * amount lost

Using our example from above of a 60% probability of a win for a -110 up bet, E(P) = 60% * 1 unit – 44% * 1 unit = 0.16 units.

Because the player was risking 1 unit, his % expected profit is just 0.16 units / 1 unit risked = 16%.

So the formula for Expected % Profit is:

E(P) = probability of win * amount won/amount lost – probability of loss


You’ll recall that in the implied winning percentages thread 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, actually the negative of theoretical hold, and has a special meaning in betting on sporting events.

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 know 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%. 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.

So in short, Calculate the theoretical, individual zero-vig implied odds, 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.

Here's another method with less steps..

Theoretical Hold = 1 – 1 / overround

Where 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%.

You may have 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.

for -105/-105 overround = 105/205+105/205 ≈ 102.44%

so the theoretical hold ≈ 1 – 1/102.44% ≈ 2.38%

for -210/+190 overround = 210/310 + 100/290 ≈ 102.22%

so the theoretical hold ≈ 1 – 1/102.22% ≈ 2.18%

for -1000/+800 overround = 1000/1100 + 100/900 ≈ 102.02%

so the theoretical hold ≈ 1 – 1 /102.02% ≈ 1.98%

So this means that the 10 cent wide spread at -105/-105

is more expensive (by about 9.4%) than the 20 cent wide spread at -210/+190

which is in turn more expensive (by about 9.9%) than the 200 cent wide spread at -1000/+800.

So contrary to what many may have thought a larger nominal spread

doesn't necessarily imply greater profitability for the book

I suppose I should be flattered that the bulk of this was lifted virtually word-for-word from my SBR post of June 2006 entitled "An introduction to expectations and theoretical hold".

See http://forum.sbrforum.com/handicappe...ical-hold.html.

No worries, but I would appreciate an attribution next time. :)

[Romanowski]

Originally Posted by Ganchrow

I suppose I should be flattered that the bulk of this was lifted virtually word-for-word from my SBR post of June 2006 entitled "An introduction to expectations and theoretical hold".

See http://forum.sbrforum.com/handicappe...ical-hold.html.

No worries, but I would appreciate an attribution next time. :)

ganch I posted it years ago, in another section, this tech section is new

I would have given you credit but generally we are not allowed to link other forums here

I always enjoy your work, and glad you came out of hiding, never see you posting at sbr anymore

in fact it would be cool if you posted here some :)

[ainokea808]

all that is great stuff but you for got to factor in the weather

[IrishTim]

Weather has nothing to do with theoretical hold. It's the same indoor or out.

[Seanie Mac]

lmao