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Maths · the arithmetic behind a price

Maths

Break-even with pushes: how a push changes the win rate you need

A push returns the stake and counts as neither a win nor a loss, so the break-even rate applies to decided bets only: wins ÷ (wins + losses). At -110 that is 52.38% of decided bets. With 8 pushes in 100 bets, 92 are decided, and 49 wins are enough, even though 49 wins in 100 looks like 49%.

The break-even calculator returns the rate for the price you enter, which you then apply to wins and losses only.

Why pushes do not change the threshold

A push is a bet whose result is exactly on the line, such as a spread of -3 in a game won by 3. The stake comes back, and the result is zero. Adding zero results to a total changes neither the winning rate among decided bets nor what that rate needs to be. Each win at -110 pays 0.909 and each loss costs 1, so you need wins × 0.909 ≥ losses, which gives wins ÷ (wins + losses) ≥ 0.5238 whatever number of pushes sits beside them.

100 bets at -110, 8 of them pushed, 1 unit each 92 are decided. At 52.38% you need 48.19 wins, so 49 whole wins. At 48 wins and 44 losses the result is 48 × 0.909 − 44 = -0.36 units. At 49 wins and 43 losses it is 49 × 0.909 − 43 = +1.55 units. The pushes sit in neither calculation.

The reporting trap

A record quoted as wins out of all bets, 49 of 100, looks like 49% and sounds like a loss. A record quoted as wins out of decided bets, 49 of 92, is 53.3%. Both are accurate and only one can be compared with 52.38%. Always say which denominator you are using. A record written as 49-43-8 (wins, losses, pushes) contains everything; the break-even test uses the first two numbers.

Return on stake

Return on stake is where pushes show up. Profit divided by the total staked changes depending on whether the pushed stakes count as staked: in the example above, +1.55 divided by 100 is 1.55% and divided by 92 is 1.68%. Neither is wrong. Pick one and keep it for the whole log, and see the bet log article for the convention.

Buying the half point

A half-point line removes the push by moving the push to a win or a loss. The price of that shift is a worse payout, for example from -110 to -120, and the break-even rate rises with it, from 52.38% to 54.55%. Whether the trade is worth it depends on how often the final margin lands exactly on the number, which is a fact about the sport and the line, not about this arithmetic. The half-point article explains the mechanics, and the key numbers article covers why some numbers matter more.

Parlays and teasers

A pushed leg in a parlay is usually removed and the ticket is paid as the shorter parlay on the remaining legs, so the stake stays in play at a lower multiple. Some teasers treat a push differently, as described in the teaser push article. The rules are in each place's terms.

Terms differ The treatment of a push in a parlay, a teaser or a prop depends on the operator and the market, and some markets have no push at all. The break-even figure here applies to a straight bet with a refunded stake. Read the terms for anything else.

Calculate it. Use the break-even calculator for the rate a price needs, then compare it with your wins divided by wins plus losses.

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