What a Push Actually Costs You on Integer Lines

There is a reason so many posted numbers end in a half. A threshold of 6.5 is a clean partition of the world: the result is either above it or below it, nothing can land on it, and settlement is never in question. The half point exists to remove an outcome, and platforms use it precisely because it makes the product simple.

Then you see the same stat posted at 6, and the removed outcome comes back. Now three things can happen. The result exceeds 6, the result falls short of 6, or the result is exactly 6. That third branch has real probability attached to it, and what the platform does with it is a rule you have to look up rather than assume.

The whole number reopens a door

Most props settle on counts: strikeouts, rebounds, shots on target, kills, takedowns, total bases. Counts are discrete. They land on whole numbers by construction, which means a whole number threshold sits directly on a value the stat can actually take. A half point threshold sits in a gap between values that nothing can ever occupy.

The industry term for landing exactly on the number is a push, and the plain fact behind it is that the probability of the exact result is not small enough to ignore. On a continuous quantity, like a distance or a time measured finely, the chance of an exact tie is effectively zero and nobody needs this article. On a stat with a handful of plausible values, one of those values is the line, and it may carry a large slice of the whole distribution. This is the discrete side of the point made in reading a distribution: the shape is made of lumps, and a whole number line puts a lump directly under the threshold.

The shortcut that quietly steals the push

Here is the specific error. Given a probability for the over, people compute the under as one minus that number. On a half point line this is correct, because the two outcomes are exhaustive. On a whole number line it is wrong, and wrong in a direction that flatters whichever side you were already leaning toward.

On an integer line the three probabilities sum to one: strictly above, exactly equal, strictly below. So the probability of the over plus the probability of the under is less than one, by exactly the weight sitting on the number itself. Subtracting the over from one does not produce the under. It produces the under plus the entire push, and if you then report that figure as your probability of winning the under, you have handed yourself the push as free profit.

This is not a rounding concern. Push weight on a low count stat can be a large share of the distribution, and an overstatement of that size will flip the sign of an expected value calculation without any other error being present. It also corrupts records after the fact: a process that treats pushes as wins in its own accounting will show a hit rate that no reader can reconcile with reality, which is one reason hit rate and projection are different claims.

Where push weight concentrates

Push probability is not spread evenly across the board. It concentrates in predictable places, and knowing them tells you when to slow down.

• Low count stats with a narrow range. If a player realistically records zero through four of something, each individual value carries a substantial share of the distribution, and a line at 2 sits on top of one of them.
• Thresholds near the center of the distribution. The most likely single value is usually near the middle, so a whole number line placed near a player's typical output lands on the heaviest lump available.
• Stats that cluster tightly. A steady contributor whose results repeat within a small band puts more weight on each value than a volatile one whose results scatter.
• Combination stats are the exception in the other direction. Adding several counts together spreads the total across many more possible values, thinning the weight on any single one.

Turn that around and a useful instinct forms. A whole number line on a wide ranging stat is close to a half point line in behavior, and you can reason about it loosely. A whole number line sitting right on a narrow ranging player's typical output is a different animal entirely, and the exact result may be the single most likely thing that happens all night.

What happens on the number is a rule, not a convention

Now the part that decides whether any of this hurts you. Different products handle the exact result differently, and the difference is not cosmetic; it changes what you bought.

In some structures the exact result voids the selection and the stake comes back. In others it is treated as a loss outright. In multiple selection entries, a single leg landing on the number may be removed with the entry recalculated at a smaller size, or it may sink the whole entry, or it may be handled differently depending on the entry type you chose. Those are three different products wearing the same posted number, and only the platform's current published rules can tell you which one you are looking at.

Because these terms vary by platform and change over time, nothing here describes anyone's current rules and you should not carry an assumption from one app to another. Check the settlement terms for the specific stat on the specific platform before the slate, not after a result lands on the number. The related mechanics of voided legs, including absences and cancellations, are covered in DNP and void rules, and the same reasoning applies to how a removed leg reshapes a multiple selection entry.

The arithmetic, with invented numbers

Nothing above requires advanced mathematics. It requires refusing to skip a term. Here is the whole thing in one worked case.

The example carries two lessons worth separating. First, the push weight was 18 points of probability that a careless calculation would have silently awarded to the under. Second, the settlement rule was worth more than most of the matchup research anyone would have done on that start. One of those is arithmetic and the other is reading, and both are faster than the analysis they protect.

The expected value framing here is the same one that governs discounted and boosted offers, where the payout moves alongside the line. If you are comfortable with modified payout lines, you already have the machinery; a push is simply a third branch in the same expected value calculation, with a payoff that is zero, or negative, depending on a rule.

Ask about the exact number before you estimate anything

The practical rule is about ordering. On any whole number line, resolve the settlement question before you form a probability, because the answer determines what you are estimating.

1. Check whether the posted threshold is a whole number at all. Half point lines need none of this.
2. Look up what the platform currently does with an exact result for that stat, in the entry type you would actually use.
3. Estimate three probabilities rather than two: above, exactly on, and below.
4. Compute the value of your side using the actual treatment of the push, and never by subtracting the over from one.
5. If the push weight is large and the rule is unfavorable, treat that as a reason to pass rather than a detail to absorb.

Slateline's engines carry push probability explicitly on integer lines rather than folding it into either side, and probabilities are computed as decision conditional: the chance of winning given that the selection settles at all, not the chance of not losing. That distinction is small in a sentence and large in a record, which is why the graded history and calibration work are published in the Model Room where the accounting can be inspected.

Half point lines let you be sloppy and never punish you for it. Whole numbers charge for the same habits, quietly, in a term nobody wrote down. When you want to compare whole number and half point versions of the same prop across platforms, Prop Grid lays the available lines side by side so the choice is visible before you commit to one.