expectations and theoretical hold

[Romanowski]

Before I talked about implied win percentages. Today I will talk about how to use those percentages to determine how this relates to a books expected profit on any given bet. This is known as theoretical hold.

The notion of expectation is central to probability and statistics.
An expectation is just an average. 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 will not always 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.

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:

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 straight up bet, E(P) = 60% * 1 unit – 40% * 1 unit = 0.2 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:

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 (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 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%. 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.

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

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.

-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 some popular opinions, larger nominal spreads don't necessarily imply greater profitability for the book.

[jaypasco]

to long to read, it looks fantastic though

[The Judge]

Good stuff here, Romo.

[Dr. Jack]

My head is spinning but good stuff Rome

[Victor]

Romanowski, great read! Can you explain it again to me please? Just kidding

[jonnyblaze]

I just decide which team will cover. Gl

[Romanowski]

Originally Posted by jonnyblaze

I just decide which team will cover. Gl

you forgot to add I have nothing to add to this thread but wanted to be included at the end of your post and before the "gl" part

[goterps74]

Originally Posted by Romanowski

you forgot to add I have nothing to add to this thread but wanted to be included at the end of your post and before the "gl" part

LOL

Man someone woke up on the wrong side of the bed today.

LOL

[Romanowski]

Originally Posted by goterps74

LOL

Man someone woke up on the wrong side of the bed today.

LOL

if my slight sarcastic comment means I woke up on wrong side of bed then what do your tyrade/rants mean?

annally raped?

anyways....anything to add? LOL

[goterps74]

Originally Posted by Romanowski

if my slight sarcastic comment means I woke up on wrong side of bed then what do your tyrade/rants mean?

annally raped?

anyways....anything to add? LOL

Sarcastic comments made by you??? LOL

Man, ur funny sometimes.

[Bobtheicon]

Good read. Thanks Romo!!

[robertg]

romo,
good stuff. How do you determine a teams probality of winning? What tools do you use? Ratings or power indexes, etc?

thanks
rg

[jonnyblaze]

Originally Posted by Romanowski

you forgot to add I have nothing to add to this thread but wanted to be included at the end of your post and before the "gl" part

Lol.

Your one to talk about posting with nothing to add. By the way Rome, really starting to lean on a NE under. Check it.

[Romanowski]

Originally Posted by robertg

romo,
good stuff. How do you determine a teams probality of winning? What tools do you use? Ratings or power indexes, etc?

thanks
rg

good question..


tons of data is my guess

[robertg]

Originally Posted by Romanowski

good question..


tons of data is my guess

i was sure of that. I just wanted to know what data among the "tons" of data out there, you use to turn the subjective into the objective. I may be over simplfying--but if you have 10 different stats our trends or factors that you like to use with a particular sport and 7 all point to the same outcome, would that be a 70% expectation of a cover?

I'm not trying to be sarcastic at all, just constantly trying to learn all that I can.
rg