Calculator

Middle Calculator

About this calculator

By · author of the MIT-licensed teachersbettextbook library ·

Most middle calculators lead with what you win if the window lands. This one leads with how often it has to.

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

A middle is two bets on opposite sides of the same game at different numbers, so a band of scores in between wins both. Land outside the band and one bet wins, one loses, and the vig is what you paid for the try.

Why that is the right number first

A middle is two bets on opposite sides at different numbers. Land between them and both win. Outside that band one wins, one loses, and you hand back the margin.

So the question is not “how much if it hits.” It is “how often does it have to hit before this stops costing money.” That number is the overround, and it is usually small, which is the entire point.

Staked for equal payouts, the middle has to land the overround of the time to break even. The form computes it from your two prices.

What it cannot tell you is how often the window actually lands. That depends on the sport and the numbers involved, and anybody offering a general answer is guessing.

Want more math?the Nerd Corner
Σ=1d1+1d2,h∗=Σ−1,P=SΣ\Sigma = \dfrac{1}{d_1} + \dfrac{1}{d_2}, \qquad h^{*} = \Sigma - 1, \qquad P = \dfrac{S}{\Sigma}

where:

  • d1,d2d_1, d_2: the decimal odds of the two sides, total return per dollar including stake. −110 is d=1.9091d = 1.9091.
  • Σ\Sigma: the sum of the two implied probabilities, dimensionless. Above 1 on a two-sided market; below 1 it is an arbitrage.
  • h∗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.
  • SS: the total stake across both sides, in dollars, split Si=S(1/di)/ΣS_i = S(1/d_i)/\Sigma so that both single-side payouts are equal.
  • PP: the payout when exactly one side wins, in dollars, identical whichever side that is.
  • hh: 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:

miss=P−S,hit=2P−S,E=h(2P−S)+(1−h)(P−S)=P(1+h)−S\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:

d1=d2=1.909091  ⇒  Σ=0.523810+0.523810=1.047619d_1 = d_2 = 1.909091 \;\Rightarrow\; \Sigma = 0.523810 + 0.523810 = 1.047619
h∗=1.047619−1=4.7619%  =  1 game in 21h^{*} = 1.047619 - 1 = 4.7619\% \;=\; \text{1 game in 21}
S1=S2=$500,P=$10001.047619=$954.55S_1 = S_2 = \$500, \qquad P = \dfrac{\$1000}{1.047619} = \$954.55
miss=$954.55−$1000=−$45.45,hit=$1909.09−$1000=+$909.09\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 has to land 1 time in 21. Not a coincidence: it is the same statement written twice, and it is the entire trade.

Why the required hit rate is exactly the overround

Split the stake so both payouts match: S1d1=S2d2=PS_1 d_1 = S_2 d_2 = P. Then S1=P/d1S_1 = P/d_1 and S2=P/d2S_2 = P/d_2, and since they add to SS:

P(1d1+1d2)=S⟹P=SΣP\left(\dfrac{1}{d_1} + \dfrac{1}{d_2}\right) = S \quad\Longrightarrow\quad P = \dfrac{S}{\Sigma}

Weighting the two nets by h and setting the total to zero:

h(2P−S)+(1−h)(P−S)=P(1+h)−S=0h(2P - S) + (1-h)(P - S) = P(1+h) - S = 0
1+h∗=SP=Σ⟹ h∗=Σ−1 1 + h^{*} = \dfrac{S}{P} = \Sigma \quad\Longrightarrow\quad \boxed{\,h^{*} = \Sigma - 1\,}

Σ−1\Sigma - 1 is the overround: the number the vig calculator prints as v and a no-vig conversion removes. So the requirement does not depend on the window, the sport or the stake, only on the prices you paid.

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

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

Two −110s: 10/21010/210, one in 21. Two −105s: 5/2055/205, one in 41. Two −102s: 2/2022/202, one in 101. The price of holding both sides is the only lever.

Push the prices far enough and Σ falls below 1: at +120 / −105 it is 0.9667 and the required rate goes negative. That is an arbitrage, the same arithmetic with Σ on the other side of 1, with the middle a bonus on top.

Reading the window

The required hit rate says nothing about the width of the window, the set of results falling strictly between the two numbers you hold:

  • +3.5 dog, −2.5 favorite: 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 prices set h∗h^{*} and a wider window is usually bought by paying worse ones: three numbers at −120 a side needs 9.09%, one number at −102 a side needs 0.99%.

Which of those clears depends on how often results land inside the window, a distribution measured on a stated sample. This page publishes none and asserts nothing about what any window is worth. It prints the bar; your estimate goes in the last field, and at exactly the required rate the panel prints $0.

One arithmetic caveat: NFL final margins cluster on 3 and 7, so equal-width windows are not interchangeable: a one-number window on 3 and one on 11 carry the same required rate and very different actual ones.

What this calculator does not model

  • Execution risk. The legs are usually placed hours apart, because a middle is created by a line move rather than found intact. Between them you hold an ordinary one-sided bet, and if the second price never appears, you still do.
  • Unequal splits. The identity assumes the equal-payout split. Weight one side more and the miss outcomes differ, so there is no single required rate: there is a surface over two probabilities.

Common mistakes

  1. 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∗h^{*}.
  2. Forgetting the vig is charged twice. A middle pays the hold on both legs, which is why h∗h^{*} is the combined overround.
  3. Taking the reverse side. Holding +2.5 and −3.5 rather than +3.5 and −2.5 turns the window into a double loss, and there is no two-sided price at which that breaks even.
  4. Treating the second leg as free insurance. Every miss costs the hold: 4.55% of the money at risk, $45.45 of every $1,000.

Frequently asked questions

What is a middle bet?

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.

What hit rate does a middle need to break even?

Exactly the combined overround: add the implied probabilities of both sides to get Σ, and the required hit rate is Σ − 1. Two −110 prices give 4.7619%, about one game in 21.

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 across two −110 prices that is $500 and $500, paying $954.55 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 a miss costs $45.45 and a hit returns $909.09.

What is the difference between a middle and an arbitrage?

The same arithmetic with Σ on the other side of 1. Below 1 both single-side outcomes are already positive and the required hit rate goes negative: that is an arbitrage, with the middle a bonus on top.

What happens if a middle lands on a push?

The pushed leg refunds its stake and the other is graded normally: across two −110 whole-number prices on $1,000, a push-and-win returns $454.55 rather than costing $45.45. Whole-number middles have five outcomes, not three.