For an event with two outcomes (fractional odds):

…you may ask yourself,

*where did this nifty equation come from? ** [see below]

DERIVATION
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

Therefore,

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

So, solving for Odds.1…

*…am I right? … am I wrong? **

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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.

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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

Therefore,

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.

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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.