Push Probability and Break-Even: How Pushes Change the Math

A push changes break-even math because it adds a third outcome that neither pays nor costs anything. At -110, a bettor needs to win 52.38% of the bets that are decided, but across every bet placed on a whole-number line the required win rate is lower, because pushes remove stake from risk. With a hypothetical 6% chance of a push, the unconditional win rate needed falls to 49.24%. The two numbers describe the same price from different angles, and mixing them up is the most common error in push math. This article treats push probability as a variable: it shows the formula for break-even with win, push and lose probabilities, works through hypothetical examples at -110, and explains why a half-point (the hook) is a price change rather than a cosmetic one. For the basic bar over a run of bets without pushes, see what the break-even win rate is; this piece covers what changes when exact ties are possible. All odds and probabilities here are hypothetical illustrations, not current market prices or measured push frequencies.

Win, Push, Lose: Three Outcomes, Not Two

A wager on a whole-number spread or total has three possible results, and their probabilities add to one: p(win) + p(push) + p(lose) = 100%. When the result lands exactly on the line, the stake is returned and nothing is won or lost. The push probability is the chance of that exact landing. On a half-point line it is zero, so the bet has only two outcomes. Every whole-number line carries a push probability above zero, even if the exact size depends on the sport and the specific number.

Two different win rates follow from this, and they answer different questions. The conditional win rate is wins divided by decided bets, which means wins plus losses with pushes ignored. The unconditional win rate is wins divided by all bets placed, pushes included in the denominator. A bettor who wins 49 of 100 bets with 6 pushes has an unconditional win rate of 49% but a conditional win rate of 49 divided by 94, or 52.13%. The conditional rate is the one that compares directly with the price, since only decided bets move money.

How a Push Changes the Break-Even Formula

Start from the price. At -110 a winning $110 bet returns $100 profit, and a losing one costs $110. Break-even means expected profit is zero, so the profit on wins must equal the loss on losses: p(win) × $100 = p(lose) × $110. A push appears on neither side of that equation, because it contributes $0. That is the whole reason the math changes: the push probability only shows up through the fact that p(win) + p(lose) = 1 − p(push).

Solving gives two forms. Among decided bets, p(win) ÷ (p(win) + p(lose)) = 110 ÷ (110 + 100) = 0.5238, or 52.38%, the same bar used for any -110 bet and identical to the ordinary implied probability formula for negative odds, |odds| ÷ (|odds| + 100). The conditional break-even rate ignores the push probability entirely. Across all bets, the required unconditional win probability is p(win) = 0.5238 × (1 − p(push)). In general, for stake S and profit P per bet, p(win) = (1 − p(push)) × S ÷ (S + P).

The same structure works for any American price. At -120 the conditional bar is 120 ÷ 220 = 54.55%, and with a 6% push probability the unconditional bar is 54.55% × 0.94 = 51.27%. At +100 the conditional bar is 50%, and with the same push probability it is 47.00%. The conditional bar depends only on price; the unconditional bar depends on price and push probability. This is also why a bettor should state which version they mean whenever they quote a required win percentage.

Worked Example: A Whole-Number Spread at -110

Suppose a sportsbook lists a hypothetical whole-number spread at -110 on both sides, and a bettor places $110 on each qualifying bet. Assume the bettor estimates that the bet pushes 6% of the time. The unconditional break-even win rate is 0.5238 × 0.94 = 49.24%, which leaves 44.76% for losses (100% − 6% − 49.24%). Check the balance: 49.24% × $100 = $49.24 of average profit, and 44.76% × $110 = $49.24 of average loss. Expected profit per bet is $0, so the price is fair at those probabilities.

The table below repeats the calculation for several hypothetical push probabilities. Each row uses the break-even condition above, so profit and loss balance exactly. The bar falls by about 0.52 percentage points for every percentage point of push probability, because it is scaled by 0.5238.

The right-hand column is the point of the table: it never changes. A higher push probability makes the all-bets figure look easier, but the standard a bettor must beat among decided bets is still 52.38%. A push lowers the number, not the difficulty. The expected vig cost also shrinks slightly: a bettor who wins exactly half of decided bets loses $5 per $110 decided, which is $4.70 per $110 ticket when 6% of tickets push (0.94 × $5).

Why Hooks and Key Numbers Matter in Price Terms

A half-point removes the push, and the three-outcome table collapses to two. Take the 6% example. On the whole-number line, the bettor wins 49.24%, pushes 6% and loses 44.76%. Move the same bet to a half-point line against the bettor (for example a favorite laying 3.5 instead of 3) at the same -110 price, and the 6% push becomes a loss: 49.24% wins and 50.76% losses. Expected profit on $110 is 49.24% × $100 − 50.76% × $110 = −$6.60, a −6.00% return on the stake. The identical price now costs much more.

That gap is the reason half-points are not priced the same as whole numbers. To be fair at 49.24% wins with no push, the price would need to be about +103, because 100 ÷ (103.09 + 100) = 49.24%. A hook changes the break-even bar from “49.24% of all bets” to “52.38% of all bets at -110”, and a buyer of the half-point must be paid, or charged, accordingly. This is the arithmetic underneath the hook and half-point pricing, and it explains why buying a half-point at the same price is not free value.

Key numbers are where this matters most. In football, certain margins occur far more often than others, so the push probability on those whole numbers is larger than on neighboring numbers, and the size of the price adjustment is larger too. The page on NFL key numbers covers why 3 and 7 behave this way. Any specific push frequency should come from historical data, not from memory, so the examples in this article use round hypothetical figures. The bigger the push probability on a number, the more a half-point should cost.

Reading a Record That Includes Pushes

Push math is most useful when applied to a results log. Take a hypothetical record of 20 wins, 18 losses and 2 pushes at -110, flat $110 stakes. Profit is 20 × $100 − 18 × $110 = $2,000 − $1,980 = +$20. The record is profitable, though barely. Among decided bets the win rate is 20 ÷ 38 = 52.63%, just above the 52.38% bar, which explains the small positive result.

Now compare how the same record looks under different bookkeeping. Counting the pushes as losses gives 20-20, a 50.00% win rate that appears to be below the 52.38% bar by a wide margin. Counting them as wins gives 22-18, a 55.00% win rate that appears comfortably above it. Neither shortcut matches the +$20 result. Only the three-way record, with its conditional rate of 52.63%, agrees with the dollars.

The unconditional comparison works too. Two pushes in 40 bets is a 5% push rate, so the unconditional bar is 52.38% × 0.95 = 49.76%, and the observed unconditional win rate of 50.00% clears it by a sliver. Return on money actually risked is $20 ÷ (38 × $110) = 0.48%; measured on all $4,400 wagered it is 0.45%. With only 40 bets, this result is well within normal variance, so it says nothing about skill. See how to track your bets for a record format that keeps pushes in their own column.

Common Mistakes With Pushes and Break-Even

The first mistake is counting a push as a loss. It lowers the apparent win rate, as in the 20-20 example, and can make a profitable run look like a failing one. The second is the mirror image: counting a push as a win inflates the record. A push is a refund, so it belongs in its own column, and tools that show only wins and losses force the bettor to track it separately.

A third mistake is quoting the unconditional bar as the standard for decided bets. Saying “I only need 49.24% at -110” is true across all bets only if the push probability really is 6%. If the actual push probability is lower, the bar is higher; if there are no pushes, it is 52.38% again. A fourth is assuming that a bet with a push is better value. Pushes reduce the vig paid, but expected profit is still zero at break-even and negative below it, so pushes are neutral, not an edge.

Where Push Math Shows Up in Practice

When a bettor compares a whole-number line with a half-point line at the same price, push math supplies the comparison. A whole-number line has a lower all-bets bar because part of the probability sits in the push; the half-point line has a higher bar because that probability has moved to a win or a loss. The two lines are different products, and the price should reflect which side of the half-point the bettor is on. Reading an odds board with that in mind makes small price differences easier to interpret.

The same arithmetic applies to totals, alternate lines and any market where ties settle as refunds. In multi-leg tickets, a pushed leg is usually removed and the ticket is repriced, which changes the break-even bar of what remains. The 6-point teaser break-even math article applies a version of this idea to teasers, where key numbers cause legs to move across push boundaries. Settlement rules decide whether a tie is a push at all, and they vary by sportsbook and market, so check the house rules for each wager.

Betting always carries risk, and understanding push math does not remove the vig or predict any single result. Use it to read records accurately and to understand prices, not as a reason to place more bets. If betting stops being enjoyable, or if tracking results begins to feel like pressure, step back and use the responsible gambling tools available.

Related Concepts and Next Steps

Push math connects pricing, records and line types. For the underlying bar without pushes, start with the break-even win rate; for what a single price represents, read how implied probability works. For how the half-point changes a wager’s price, the hook and half-point guide goes deeper. The next logical step is expected value, which turns a bettor’s own probability estimate, including a push probability, into an expected result per bet. Keep each estimate flagged as an estimate, and treat the formulas as a way to ask better questions rather than a source of answers.