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 astheoretical hold.

The notion ofexpectationis central to probability and statistics. Essentially an expectation is just an average with an extra syllable. If you flip a coin 10 times then youexpectit will land on heads 5 times and youexpectit 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 thenon averageit 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: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(Unit Profit) = probability of win * amount won – probability of loss * amount lost^{1}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 asCode:E(P) = probability of win * amount won/amount lost – probability of losstheoretical 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 bookalwaysexpects 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 thatoverroundis 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:

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.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%

- 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) - … .
- 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.