Nonlinear odds-to-probs conversion

  • The New England Patriots are playing the Buffalo Bills.
  • The fractional odds are Patriots 1/2 and the Bills 2/1, where a bet on the Patriots risks $2 to win $1, and a bet on the Bills risks $1 to win $2.
  • A rational bettor has $1 to bet.
  • A $1 wager on the favorite Patriots yields a $0.50 profit, if they win.
  • Or, a $1 wager on the underdog Bills yields a $2.00 profit, if they win.
  • Therefore, the relative payout on the Bills is 4X that of the Patriots (2.00/0.50; this simple ratio mathematically ties the teams’ odds together, making the odds-to-probability relationship nonlinear).
  • As odds-and-probs have an inverse relationship, the probability that the Bills win is 1/4th that of the Patriots.
  • Therefore, the Bills have a 20% chance of winning, which is 1/4th that of the Patriots 80% chance of winning.
Can you show that this is NOT the case?

The conventional conversion of odds-to-probability is inverse-linear, where the probabilities are derived independently for each outcome from its odds. This overstates the probability of the underdog, while understating the probability of the favorite ... hence, the well-known 'longshot bias'.
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This chart shows the odds-to-probs 'misunderstanding' -- assuming the above is correct -- of the implied probability from the underdog's odds (the difference between linear-inverse and nonlinear-inverse conversion).

As noted earlier, this could account for 'longshot bias' in betting.

Note: The absolute misunderstanding -- the difference between linear and nonlinear conversions -- appears to be shrinking as the underdog's odds increase, but the relative error 'misunderstanding' goes thru the roof.
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Assuming that 'odds' is the fractional odds for an outcome (or the payout on a $1 bet), the odds and its implied probability have a direct inverse-proportional relationship: Prob.x = f(1/Odds.x) and Odds.x = f(1/Prob.x).

Therefore, with any zero-sum event with n competitive outcomes:

Odds.1 x Prob.1 = Odds.2 x Prob.2 = ... = Odds.n x Prob.n

In the example of the hypothetical football game:

Patriots: 1/2 x 0.8 = 0.4​
Bills: 2/1 x 0.2 = 0.4​

Using the conventional inverse-linear odds-to-probs conversion -- where 1) Prob.x = f[1/(Odds.x+1)], and then 2) normalizing the overround -- distorts this natural relationship between odds and probability.

As a test of functionality of odds-to-probs conversion, take listed odds in a zero-sum event and convert them to implied probabilities in each of the outcomes. Then, take those calculated probabilities and convert them back into their implied odds (no house take). How do those implied odds compare to the listed odds?
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On the conversion of probabilities into their implied odds, the following relationship exists for a game between Team 1 and Team 2:


Therefore, using the above NFL example:


QED ... patents pending, for two-or-more event outcomes.
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There was no response to my odds-to-probs-to-odds challenge (above, 3rd post)...
As a test of functionality of odds-to-probs conversion, take listed odds in a zero-sum event and convert them to implied probabilities in each of the outcomes. Then, take those calculated probabilities and convert them back into their implied odds (no house take). How do those implied odds compare to the listed odds? I took the liberty to answer my own question, using the earlier NFL example.

That's wtf ... and thanks for asking, I wouldn't want to be unclear.
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For an event with two outcomes (fractional odds):

…you may ask yourself, where did this nifty equation come from? * [see below]


For a two-outcome event (outcomes #1 or #2) with no house take, the fractional odds of an outcome is the square-root of the relative probability of the opposite outcome … for instance, a 50/50 chance, such as a coin flip, would have a relative probability of 1, and therefore odds = 1, for both heads and tails. The derivation of this relationship is provided below.

In a settled wager, the amount risked, or bet, on an outcome is the same as the profit from the bet on the opposite outcome. The bettors just compensate each other after the event, based on whose choice won. In the earlier NFL example, a $2 bet on the Patriots results in a $1 profit if the Patriots win, while a $1 bet on the Bills results in a $2 profit if the Bills win. There’s $3 in the pot with combining both bets ($2 + $1) … who wins the pot depends on the game result.

The inherent bet symmetry can be summarized for a two-outcome event, with no house take:

Bet.1 = Profit.2​
Bet.2 = Profit.1​

In addition, the definition of fractional odds on the outcomes is:

Odds.1 = Profit.1/Bet.1​
Odds.2 = Profit.2/Bet.2​

So, substituting for Profit.x in each fractional-odds equations…

Odds.1 = Bet.2/Bet.1​
Odds.2 = Bet.1/Bet.2 = 1/Odds.1

And we already know (see post #3)…

Odds.1 x Prob.1 = Odds.2 x Prob.2​
Odds.1 x Prob.1 = (1/Odds.1) x Prob.2​

So, solving for Odds.1…


…am I right? … am I wrong? *


--------[ WAIT ... THERE'S MORE!!! ]------

DISCLOSURE (on lack thereof):

On later TalkStats threads, odds in zero-sum events have been used to determine the implied probabilities of outcomes, in both sports and politics ... readers might look at those numbers and wonder "How did I get here?"

Well, I had left out a piece of the puzzle, since when posts were written, the US patent on the algorithmic technology of nonlinear odds-to-probs conversion had not yet been awarded. The USPTO has since awarded and is expected to publish the patent shortly, so what the heck…

It is known, and has already been disclosed, that odds and probability of n number outcomes in zero-sum events have a monatomic nature:
Odds.1 x Prob.1 = Odds.2 x Prob.2 = …Odds.n x Prob.n = Constant
What’s not known, or at least not as widely as it should be, is that the Odds alone have another monatomic relationship ... with each other. For a two-outcome zero-sum event (with no house take), the following is true:
Odds.1 x Odds.2 = 1 (Payout Product)
The Patriots/Bills example above provides a good example. The Odds.Patriots (1/2) and Odds.Bills (2/1) are just the inverse of each other, duh. Fractional and American odds show this relationship … on the other hand, decimal odds – the most popular format, worldwide – obscure the relationship, but who cares since we now have massive computing power to really obscure functional relationships and reduce understanding, on an industrial scale.
Betting houses are in business to make money, so this Payout Product (‘Payout’ is the same value as ‘Odds’, the decimal equivalent of fractional odds) will be less than 1.00 for a two-outcome event, generally in the range of 0.8-0.95 for widely followed, popular sporting events, but considerably less for obscure or uncertain events with little coverage, where the house fears overweight wagering on one side of the line, or just wants to profit from eager/naïve bettors.
Those two monatomic relationships are actually all that is needed to derive the now-patented nonlinear odds-to-probs algorithm. The methodology is very simple math (ratios, inverses) that gets a bit funky, but then reduces to something elegant.

Care to give it a shot?

If so, do the derivation on a two-outcome event, and the same algorithmic structure -- reduced, properly -- extends to multiple outcomes as well.

I had been doing the two-outcome odds-to-probs conversion for almost 20 years on my calculator watch, but didn't apply for the patent until I figured out how to do it for three-or-more outcomes, since it then required a computer for efficient calculations, which is a critical condition of patentability of algorithms, which sounds reasonable.

--------[ WAIT ... THERE'S MORE!!! ]-------

DERIVATION (two-outcome event, baby steps):

The simple-but-funky mathletics performed in the initial post can be represented by the formulas:


To build a generic nonlinear odds-to-probs model, we need to directly relate the odds of the two competing outcomes, like with Patriots’ 1/2 and the Bills' 2/1, but now with a formula. To do this, we will assume an algorithmic relationship exists across the range of odds spectrums for two-outcome events, from ‘even odds’ (equally probable outcomes) to disparate odds (heavy favorite with longshot underdog) between competitive outcomes.

Let’s assume that, for a game between Team.X and Team Y that the product of the Odds.X and Odds.Y is a constant Z:

Odds.X x Odds.Y = Z​

As this function must be continuous across the range of all odds-combinations, it must hold at even odds (1/1), as with flipping a coin, where one expects that a $1 bet risked on heads or tails yields a $1 profit, if correct. So, the equation becomes:

Z = Odds.X x Odds.Y = 1/1 x 1/1 = 1​


Odds.Patriots x Odds.Bills = 1​

This aligns with the above Patriots-vs-Bills odds example, where 1/2 x 2/1 = 1.

In real-life betting markets with two-outcome events, the Odds-Product Z for a two-outcome event is less than 1.00, often by a substantial margin, due to the betting house take (t). Therefore, the model formula would become:

Odds.Patriots x Odds.Bills = 1 - t

But, in the interest of clarity and simplicity, we will assume that the house take is zero, and so in a game between Team.X and Team.Y:

Odds.X x Odds.Y = 1​
Odds.X = 1/Odds.Y​

And, so with the relationship between Odds.X and Odds.Y now established for a two-outcome event, X or Y...


One of the above formulas brushes against the generic nonlinear odds-to-probs algorithm for two-or-more outcomes in a zero-sum event.

Please note that this nonlinear odds-probs functionality can be utilized for building efficient simple betting markets. Several patents are pending on said nonlinear betting markets, as well as on 'running this functionality backwards' and performing the nonlinear probs-to-odds conversion, as was demonstrated initially in this post.

--------[ WAIT ... THERE'S MORE!!! ]-------

This is the above equation that can be utilized to derive a generic algorithm for odds-to-probs conversion for a zero-sum event:


The other equations, including the monatomic Odds.X x Odds.Y = 1, were required to derive the chart on the 2nd post, which compares the odds-vs-probability of the linear and nonlinear conversion methods.
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On the nonlinear conversion of odds to implied probability, the following applies for two outcomes in a zero-sum competition:


In this functional relationship, the two competitive odds are considered independently and then are combined, as would be done for a logarithmic average, for determining relative implied probability. A heavy favorite has low odds (<<1), so its odds' impact as an inverse on implied probability will be commensurately weighty, hence the term 'heavy'.
The odds-to-probability relationship for two outcomes (shown above) can be expanded to multiple outcomes, and expressed in a generic algorithm for n number of outcomes in a zero-sum event ('Odds' are fractional odds, which can be expressed as a decimal):


This algorithm creates the condition where the odds and probability of a given outcome are monatomic at all times, as they should be, and so for any given zero-sum event:

Odds.1 x Prob.1 = Odds.2 x Prob.2 = ... = Odds.i x Prob.i …= Odds.n x Prob.n = Constant

which is a handy way to determine whether the patented nonlinear odds-to-probability conversion has been utilized.

--------[ WAIT ... THERE'S MORE!!! ]------


If you are confronted by linearly-derived, odds-implied probabilities for a two-outcome event, there is an easy hack that will closely approximate the nonlinear odds-to-probs conversion. We have seen that...


Due to the normalizing of linear odds-to-probs conversions generally performed to ensure that the outcomes’ probabilities sum towards 100%, the hacked ‘corrected probabilities’ will not be exactly the same as the nonlinear conversions, but very close.

Of course, the existence of other outcomes in addition to two outcomes -- often a minor possibility, such as a third party in American politics -- could change their overall probabilities downward somewhat, but will not impact the probability relationship between the two main outcomes.


If you are interested in utilizing this patented odds-to-probs technology commercially, please contact

NOTE: All patent protections will be enforced to the fullest extent of the law. Anyone reporting actionable commercial violations to the patent holder will be compensated with a percentage of damages won. Contact for details.
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