Middle Calculator

By Alex Thornton · author of the MIT-licensed teachersbettextbook library · Last reviewed 24 July 2026

Most middle calculators lead with the payout if the window lands. That is the least informative number on the page. This one leads with the required hit rate — how often the window has to come in before the position stops costing money.

A middle is two bets on opposite sides of the same game at different numbers, so a band of results wins both. Outside that band one bet wins and the other loses, and the position gives back the vig. Staked for equal payouts, the middle has to land Σ − 1 of the time to break even.

\Sigma = \dfrac{1}{d_1} + \dfrac{1}{d_2}, \qquad h^{*} = \Sigma - 1, \qquad P = \dfrac{S}{\Sigma}

where:

$d_1, d_2$ — the decimal odds of the two sides, total return per dollar including stake. American −110 is $d = 1.9091$ .
$\Sigma$ — the sum of the two implied probabilities, dimensionless. Above 1 on a normal two-sided market; below 1 it is an arbitrage.
$h^{*}$ — the required hit rate: how often the window must land for the position to return exactly zero. It equals the overround on the two prices, which is the whole point of this page.
$S$ — the total stake across both sides, in dollars, split $S_i = S(1/d_i)/\Sigma$ so that both single-side payouts are equal.
$P$ — the payout when exactly one side wins, in dollars — identical whichever side that is.
$h$ — your estimate of how often the window lands, between 0 and 1. Nothing on this page supplies it.

The three outcomes, and the expected value across them:

\text{miss} = P - S, \qquad \text{hit} = 2P - S, \qquad \mathbb{E} = h(2P - S) + (1-h)(P - S) = P(1 + h) - S

Worked example — the numbers in the form below

Two −110 prices on opposite sides of the same game at different numbers, $1,000 total:

d_1 = d_2 = 1.909091 \;\Rightarrow\; \Sigma = 0.523810 + 0.523810 = 1.047619

h^{*} = 1.047619 - 1 = 4.7619\% \;=\; \text{1 game in 21}

S_1 = S_2 = \$500, \qquad P = \dfrac{\$1000}{1.047619} = \$954.55

\text{miss} = \$954.55 - \$1000 = -\$45.45, \qquad \text{hit} = \$1909.09 - \$1000 = +\$909.09

The window pays 20 times what a miss costs, and it has to land 1 time in 21. That is not a coincidence — it is the same statement written twice, and it is the entire trade. Everything else on a middle slip is decoration.

When the same two prices sum below 1 — the arbitrage calculator →

Why the required hit rate is exactly the overround

This falls out in four lines and it is the most useful identity on the page, so here it is in full.

Split the stake so both single-side payouts match: $S_1 d_1 = S_2 d_2 = P$ . Then $S_1 = P/d_1$ and $S_2 = P/d_2$ , and since they add to $S$ :

P\left(\dfrac{1}{d_1} + \dfrac{1}{d_2}\right) = S \quad\Longrightarrow\quad P = \dfrac{S}{\Sigma}

Exactly one side winning returns $P$ , so the net is $P - S$ . Both sides winning returns $2P$ , so the net is $2P - S$ . Weight them by $h$ and set the total to zero:

h(2P - S) + (1-h)(P - S) = P(1+h) - S = 0

1 + h^{*} = \dfrac{S}{P} = \Sigma \quad\Longrightarrow\quad \boxed{\,h^{*} = \Sigma - 1\,}

$\Sigma - 1$ is the overround: the same number the vig calculator prints as v, and the same number a no-vig conversion removes. (The hold is its smaller sibling, v / (1 + v) — 4.55% on two −110s — and that is what a miss costs as a share of the money risked.) So a middle’s break-even requirement does not depend on the size of the window, the sport, the stake, or which numbers you caught. It depends on the prices you paid, and on nothing else.

For two identical prices at − $x$ the whole thing collapses to a fraction you can do in your head:

h^{*} = \dfrac{x - 100}{x + 100} \qquad \Rightarrow \qquad \text{1 in } \dfrac{x + 100}{x - 100}

Two −110s: $10/210$ , one in 21. Two −105s: $5/205$ , one in 41. Two −102s: $2/202$ , one in 101. The price you pay to hold both sides is the only lever on this number, and it moves it enormously.

Prices	Σ	Required hit rate	About	Miss (on $1,000)	Hit
−102 / −102	1.0099	0.990%	1 in 101	−$9.80	+$980.39
−105 / −105	1.0244	2.439%	1 in 41	−$23.81	+$952.38
+105 / −125	1.0434	4.336%	1 in 23	−$41.56	+$916.88
+100 / −120	1.0455	4.545%	1 in 22	−$43.48	+$913.04
−110 / −110	1.0476	4.762%	1 in 21	−$45.45	+$909.09
−120 / −110	1.0693	6.926%	1 in 14	−$64.78	+$870.45
+120 / −105	0.9667	−3.326%	not required	+$34.40	+$1,068.81

The last row is the boundary case. At +120 / −105 the two prices sum below 1, so both single-side outcomes are already positive and the required hit rate is negative — the position needs the window zero times. That is an arbitrage, and the middle is a bonus sitting on top of it. Middles and arbs are not different techniques; they are the same arithmetic with Σ on either side of 1.

Reading the window

The required hit rate says nothing about how wide the window is, which is a separate question with a separate answer: the window is the set of results that fall strictly between the two numbers you hold.

+3.5 on the underdog and −2.5 on the favorite — the window is a favorite win by exactly 3. One number.
+3.5 and −1.5 — margins of 2 or 3. Two numbers.
Over 44.5 and Under 47.5 — totals of 45, 46 or 47. Three numbers.

Wider is not automatically better, because the two prices are what set $h^{*}$ and a wider window is usually bought by paying worse ones. A three-number window at −120 on both sides needs 9.09% (one in 11); a one-number window at −102 on both sides needs 0.99% (one in 101). Which of those clears is a question about how often the specific results inside the window occur — and that comes from a distribution of final margins or totals, measured on a stated sample. This page does not publish one and does not assert what any window is worth. It prints the bar; the estimate is yours, and it goes in the last field.

One structural note that is arithmetic: in the NFL, final margins cluster on 3 and 7 far more heavily than on any other number, so windows are not interchangeable at equal width. A one-number window on 3 and a one-number window on 11 have the same required hit rate and very different actual ones. The required hit rate is a property of the prices; the realized rate is a property of the sport.

Worked example: unequal prices, +105 and −125

The dog side was caught at +105 before the line moved; the other side is available at −125. Total stake $1,000.

d_1 = 2.05,\quad d_2 = 1.80 \;\Rightarrow\; \Sigma = 0.487805 + 0.555556 = 1.043360

S_1 = \$1000 \times \dfrac{0.487805}{1.043360} = \$467.53, \qquad S_2 = \$532.47

P = \$467.53 \times 2.05 = \$532.47 \times 1.80 = \$958.44

h^{*} = 4.3360\%, \qquad \text{miss} = -\$41.56, \qquad \text{hit} = +\$916.88

The requirement fell from 4.76% to 4.34% — 9% off the bar — even though the second side got worse. Catching +105 instead of −110 cuts 3.60 points off Σ; paying −125 instead of −110 adds 3.17 back. The 15-cent gain on one leg more than covers the 15-cent loss on the other, because implied probability is not linear in the American price. The stake split does the equalizing work: $467.53 at 2.05 and $532.47 at 1.80 both return $958.44, so a miss costs the same whichever side wins.

Pushes are a third outcome, and they are good

The formula above has two outcomes per leg. Middles built on whole numbers have three, and the extra one lands in your favor: a pushed leg refunds its stake and the other leg is graded normally.

A position holding Over 45 and Under 47 at −110 each, $500 a side: a total of 46 wins both. A total of exactly 45 pushes the over and wins the under:

\text{push + win} = S_2 (d_2 - 1) = \$500 \times 0.909091 = +\$454.55

That is not a miss at −$45.45; it is a return of $454.55. Whole-number middles therefore have five outcomes — win both, push-and-win either way, or miss either way — rather than the three a half-point middle has, and the two-payout break-even formula understates them. Half-point middles (44.5 / 47.5) have no push and the formula on this page applies exactly.

The reverse middle: when both sides can lose

Reverse the numbers — take +2.5 on the dog and −3.5 on the favorite — and a margin of exactly 3 loses both. Call that probability $g$ :

\mathbb{E} = (1-g)(P - S) + g(-S) = P(1-g) - S

P(1-g) = S \;\Longrightarrow\; g = 1 - \Sigma

With

\Sigma > 1

that break-even

g

is negative, and a probability cannot be negative. So a reverse middle at ordinary two-sided prices has no break-even point at all: it loses

P - S

at best and the whole stake when the gap lands. On the default −110 / −110 prices that is −$45.45 at best and −$1,000 at worst. This is the same arithmetic as the middle, read in the direction most people do not check.

What this calculator does not model

Execution risk. The two legs are usually placed minutes or hours apart, because a middle is normally created by a line move rather than found intact. Between the legs you are holding an ordinary one-sided bet, and if the second price never appears you are still holding it.
Correlated grading. Spread and total middles across different books can be graded on different official sources. Rare, and expensive when it happens.
Limits and account life. Both sides have to actually fill at the size the split calls for.
Unequal splits. The required-hit-rate identity assumes the equal-payout split. Weight one side more heavily and the two miss outcomes stop being equal, so there is no single required rate any more — there is a surface over two probabilities.

When to use this calculator

After a line moves through a number you already hold. That is where middles come from; this page prices the position the move created.
To compare two candidate middles. The one with the wider window is not the cheaper one — compare $h^{*}$ , which is set by the prices alone.
To size the second leg. The split field returns the stake that equalizes the two miss outcomes, so the downside is one number instead of two.
To test your own frequency estimate. Enter it and the panel prints the expected value it implies, and prints $0 when the estimate sits exactly on the required rate.

Common mistakes

Leading with the payout. +$909.09 on a hit is the same fact as −$45.45 on a miss, and neither is informative without $h^{*}$ .
Assuming a wider window is a better position. Window width and required hit rate are set by different things — the numbers and the prices respectively.
Forgetting the vig is charged twice. A middle pays the hold on both legs, which is why $h^{*}$ is the combined overround rather than a single book’s margin.
Taking the reverse side. Holding +2.5 and −3.5 rather than +3.5 and −2.5 turns the window from a double win into a double loss, and there is no price at which that breaks even.
Ignoring push rules on whole numbers. They move the arithmetic in your favor, and leaving them out understates the position.
Treating the second leg as free insurance. Every miss costs the hold. A middle that never lands bleeds 4.55% of the money at risk — $45.45 of every $1,000 — one game at a time.

Check this number yourself

The stake split is the same 1/d weighting the MIT-licensed teachersbettextbook package uses for arbitrage, and the required hit rate is one subtraction on top of it:

pip install teachersbettextbook

from teachersbettextbook import moneyline_to_odds

S = 1000.0
d1 = 1 + moneyline_to_odds(-110)      # 1.9090909...
d2 = 1 + moneyline_to_odds(-110)
sigma = 1 / d1 + 1 / d2               # 1.0476190...
h_star = sigma - 1                    # 0.0476190  -> 1 in 21
s1, s2 = S * (1 / d1) / sigma, S * (1 / d2) / sigma   # 500.0, 500.0
P    = S / sigma                      # 954.5454...
miss, hit = P - S, 2 * P - S          # -45.4545, +909.0909

Same inputs, same digits, no account required. More on what is open and what is not: open source.

A middle and an arbitrage are the same calculation on either side of Σ = 1, and the quantity that separates them — the overround — is what the vig calculator prints and what the no-vig calculator removes. To close a position rather than widen one, see the hedge calculator. For the other two multi-leg structures on a bet slip, round robin and teaser. And Arbitrage Betting covers the execution problems that no stake split can solve.

Frequently asked questions

What is a middle bet?

What hit rate does a middle need to break even?

Exactly the combined overround on the two prices. Add the implied probabilities of both sides to get Σ; the required hit rate is Σ − 1. Two −110 prices give Σ = 1.047619, so the window has to land 4.7619% of the time — about one game in 21. At −105 on both sides that falls to 2.439%, one in 41.

How do you split the stakes on a middle?

In proportion to 1/d on each side, the same weighting an arbitrage uses, which makes both single-side payouts identical. On $1,000 at −110 and −110 that is $500 and $500, paying $954.55 either way. At +105 and −125 it is $467.53 and $532.47, paying $958.44 either way.

How much do you lose when a middle misses?

The single-side payout minus the total stake, which is the hold on the two prices. On $1,000 across two −110 prices the miss costs $45.45 and the hit returns $909.09. That ratio is the whole trade: the window pays 20 times what a miss costs, and it has to land once in 21 to break even.

What is the difference between a middle and an arbitrage?

The same arithmetic with Σ on the other side of 1. When Σ is below 1 both single-side outcomes are already positive and the required hit rate goes negative — that is an arbitrage, and the middle is a bonus on top. When Σ is above 1 every miss costs money and only the window makes it back.

What happens if a middle lands on a push?

The pushed leg refunds its stake and the other leg is graded normally, which is much better than a miss. On $1,000 split evenly across two −110 whole-number prices, a result that pushes one leg and wins the other returns $454.55 rather than costing $45.45. Whole-number middles therefore have five outcomes — win both, push-and-win either way, or miss either way — rather than the three a half-point middle has, and the two-payout break-even formula understates them.