Arbitrage Betting: The Math Behind the Sure Bet

By · author of the MIT-licensed teachersbettextbook library ·

Two boards, one race, and a gap wide enough that both sides win. Finding it is the easy half.

Meet your classmates

Same five, and today there are two boards.

Two boards, and one of them is late

Same two hamsters. Chalk against Pip, still a coin flip, still nobody asking them how they feel about it.

What is new is that there are two boards. Your class piled on Pip again, and the board across the road moved: Chalk out to +112, Pip in to −124.

Bookie Brian has not touched his board since 11:41. He still has the race at −105 either way.

Both boards still take a vig. Brian keeps 2.44%, the other board 2.53%. A window opens only when two boards disagree by more than those two slices put together. Which is why there is almost never one.

Add the two percentages up

Turn the best price on each hamster into the probability it implies. Chalk at +112 is 47.17%. Pip at −105 is 51.22%. Add them: 98.39%.

Under 100% means you can buy both outcomes for less than either one pays back. One ticket has to win, and it wins by more than the pair of them cost. That is an arbitrage.

Over 100% there is nothing there. That is the answer nearly every time you look.

Then split the stake so both tickets return the same amount. $51.25 on Chalk, $55.65 on Pip. Either way $108.65 comes back on $106.90 out.

Most of the day there is nothing there

Here is one game across a made-up trading day. The line is the cost of covering both sides. Above 1.00 there is no lock, and that is where it sits nearly all day.

It dips under when one board naps, and it comes straight back once that board wakes. Measured on real books, many arbs die within minutes of opening.[3]

Synthetic day: one game's true probability drifts as news arrives; each book reprices with its own vig and its own laziness, so gaps open when one sleeps. Tuned so nearly every run shows a window or two (real windows are rarer, smaller, and close faster). Execution risk is the part no chart shows: two tickets at two books never fill in the same instant, and a line that moves between your legs turns a lock into a plain bet. An illustration, not an inventory.

See live arbs on the Live Board →Free. Arbs surface with the stake split and execution warnings.

Watch the box close. Then watch it not.

Evan takes Chalk at the board that moved, then turns to Brian for Pip. The first ticket is easy. The second one is a race, and he does not get to run it twice.

Bookie Brian +EV Evan Brian’s board 11:41 Chalk −105 Pip −105 12:05 −124 The other board 12:04 Chalk +112 Pip −124 LEG 1 $51.25 on Chalk +112 returns $108.65 LEG 2 $55.65 on Pip −105 returns $108.65 LEG 2 Not filled the board moved first 47.17% + 51.22% = 98.39% $106.90 out. $108.65 back, whoever wins. Locked profit $1.75, or 1.64%. 47.17% + 55.36% = 102.53% One leg on. $51.25 live on a coin flip. He was playing for $1.75. Now the swing is $108.65. Two boards, one race, and Brian is 23 minutes behind. Evan takes Chalk at the other board, then turns to Brian for Pip. $106.90 across two counters. $108.65 back either way. Brian wakes up between the legs. One ticket on, nothing to hedge it.
Arithmetic, not a drawing. Every figure here comes out of the four prices on the two boards. Bookie Brian holds 2.44% on his market and the other board holds 2.53% on its own, so neither one is giving anything away by itself. The window exists only because they disagree about Chalk by more than those two margins put together, and it shuts the moment Brian catches up. The stakes are the split that forces both tickets to return the same amount, which is why $51.25 and $55.65 both come back as $108.65.

$106.90 across two counters, to make $1.75. Miss one leg and you are holding $51.25 on a coin flip, chasing a dollar seventy five.

Here is the part that took me a while to see. The leg you keep is usually the good price, because a good price is why the gap was there.

Chalk at +112 on a coin flip is a fine number. It is just a $51.25 bet, placed by somebody trying to have no bet at all.

So the arithmetic is the easy part. Landing both tickets is the job. The second board can reject you, cut your limit, or void the leg after it settles, and a gap you cannot execute is not a gap.

One more. Anything paying much over 5% is almost always a stale price nobody has taken down. Read a big one as a warning.

This site flags an arbitrage on the board and posts it to a Discord channel, stamped with both timestamps. It will never push one to your phone. A window that lives seconds cannot outrun a cron job.

Want more math?the Nerd Corner

Nerd Corner

Advanced material. Nothing above depends on it.

A locked arbitrage, end to end

Worked example: a locked arbitrage

Book A, side X: +105  (decimal 2.05)\text{Book A, side X:}\ +105\ \ (\text{decimal } 2.05)
Book B, side Y: +110  (decimal 2.10)\text{Book B, side Y:}\ +110\ \ (\text{decimal } 2.10)
12.05+12.10=0.4878+0.4762=0.9640<1  ⇒  arb\dfrac{1}{2.05} + \dfrac{1}{2.10} = 0.4878 + 0.4762 = 0.9640 < 1 \;\Rightarrow\; \text{arb}
Return: 10.9640−1=3.7%⋅stake≈$506 / $494 per $1,000\text{Return: } \dfrac{1}{0.9640} - 1 = 3.7\% \quad\cdot\quad \text{stake} \approx \$506\,/\,\$494 \text{ per } \$1{,}000
Both outcomes pay back about $1,037 on a $1,000 outlay: a locked 3.7%, if you beat the clock.

The lock, derived: when two books disagree enough

Cover both sides at two books, force the payouts equal, and the lock condition ∑1/d<1\sum 1/d < 1 falls straight out. The whole argument, board-style:

The general n-outcome arbitrage

Nothing in the machinery counts to two. Any market with nn mutually exclusive outcomes works identically: take the best decimal price on each outcome and sum the inverses. That sum, SS, is the whole story.

The condition, for any n

S=∑1di=1d1+1d2+⋯+1dnS = \sum \dfrac{1}{d_i} = \dfrac{1}{d_1} + \dfrac{1}{d_2} + \cdots + \dfrac{1}{d_n}
Arb exists when S<1\text{Arb exists when } S < 1
S is the cost of guaranteeing a $1 return across all n outcomes. Under $1, same test as two-way, just more terms.

The stake split is the only split that survives one requirement: identical payout whichever outcome wins. Impose that and the algebra does the rest in two lines.

The stake split, derived

Equal payouts: si di=P for every outcome i, so si=Pdi\text{Equal payouts: } s_i\,d_i = P \text{ for every outcome } i, \text{ so } s_i = \dfrac{P}{d_i}
T=s1+⋯+sn=P(1d1+⋯+1dn)=P S  ⇒  P=TS,si=Tdi ST = s_1 + \cdots + s_n = P\Big(\dfrac{1}{d_1} + \cdots + \dfrac{1}{d_n}\Big) = P\,S \;\Rightarrow\; P = \dfrac{T}{S},\quad s_i = \dfrac{T}{d_i\,S}
Line one forces stakes proportional to inverse odds; line two pins the constant with total T. The prices fully determine the split.

With the guaranteed payout P=T/SP = T/S in hand, the return:

Guaranteed return

r=PT−1=1S−1r = \dfrac{P}{T} - 1 = \dfrac{1}{S} - 1
Positive exactly when S<1S < 1. The worked example’s 3.7% above is this formula with n=2n = 2: 1/0.9640−11/0.9640 - 1.

The n-outcome form earns its keep on 3-way markets: soccer’s 1X2, regulation-time hockey. Three books each shading a different outcome can open a gap, but three legs mean three fills before any line moves. One trap: all nn prices must settle on the same terms. A “draw no bet” leg refunds on the draw; counting it as draw cover breaks the lock silently.

Worked example: a three-way lock

Book A, Home: 2.30⋅Book B, Draw: 3.20⋅Book C, Away: 4.10\text{Book A, Home: } 2.30 \quad\cdot\quad \text{Book B, Draw: } 3.20 \quad\cdot\quad \text{Book C, Away: } 4.10
S=12.30+13.20+14.10=0.4348+0.3125+0.2439=0.9912<1  ⇒  arbS = \dfrac{1}{2.30} + \dfrac{1}{3.20} + \dfrac{1}{4.10} = 0.4348 + 0.3125 + 0.2439 = 0.9912 < 1 \;\Rightarrow\; \text{arb}
Stakes per $1,000: Tdi S  ⇒  $439 / $315 / $246⋅payout TS=$1,009\text{Stakes per } \$1{,}000:\ \dfrac{T}{d_i\,S} \;\Rightarrow\; \$439\,/\,\$315\,/\,\$246 \quad\cdot\quad \text{payout } \dfrac{T}{S} = \$1{,}009
r=10.9912−1=0.9%r = \dfrac{1}{0.9912} - 1 = 0.9\%
Home, draw, or away, the tickets return about $1,009 on a $1,000 outlay: a locked 0.9%, if all three legs fill at the shown prices.

An arb needs two books that disagree; a positive-expected-value bet needs only one book that’s wrong (see expected-value betting). Or run any prices through the free arbitrage calculator for the condition and the split.

Where the sure bet comes from

The condition predates sportsbooks. Prices whose implied probabilities sum away from one are what probability theory calls incoherent, and incoherent prices are exactly the ones a bettor can lock a sure win against.[1][2] ∑1/d<1\sum 1/d < 1 is that argument read from the bettor’s side of the counter. The warnings above are measured record, not folklore: arbs across bookmakers are real but limited,[4] cross-venue locks are documented at scale,[5] and the market cleans up mispricings fast.[3]

  1. Ramsey, F. P. (1931, written 1926). “Truth and Probability.” In R. B. Braithwaite (ed.), The Foundations of Mathematics and Other Logical Essays. London: Kegan Paul, Trench, Trubner & Co., 156–198. – the first “book” argument: incoherent beliefs can be bet against for a sure loss.
  2. de Finetti, B. (1937). “La prévision: ses lois logiques, ses sources subjectives.” Annales de l’Institut Henri Poincaré 7(1), 1–68. Numdam. – coherence: prices summing away from one admit a sure win.
  3. Marshall, B. R. (2009). “How Quickly Is Temporary Market Inefficiency Removed?” The Quarterly Review of Economics and Finance 49(3), 917–930. – the decay clock: many arbs die within minutes, most inside an hour.
  4. Vlastakis, N., Dotsis, G. & Markellos, R. N. (2009). “How Efficient Is the European Football Betting Market? Evidence from Arbitrage and Trading Strategies.” Journal of Forecasting 28(5), 426–444. – inter-bookmaker arbs measured: real, profitable, limited.
  5. Franck, E., Verbeek, E. & Nüesch, S. (2013). “Inter-market Arbitrage in Betting.” Economica 80(318), 300–325. – book-vs-exchange locks at scale in top-league soccer.

Every formula here lives on the Formula Sheet for quick reference.

Check your understanding

Frequently asked questions

What is arbitrage betting?

Backing every outcome of a market across different books at prices generous enough to guarantee a profit regardless of the result.

How do you know if an arbitrage exists?

Convert each side's best price to decimal odds and sum the inverses. If 1/d₁ + 1/d₂ is under one, an arb exists; one minus that sum is the locked margin.

How do you split stakes for an arb?

Stake each side in proportion to its inverse decimal odds so every outcome returns the same amount.

Is arbitrage betting risk-free?

The math locks only if both legs fill at the shown prices. Lines move in seconds, books limit arbitrageurs, and a voided leg breaks the arb.