Odds, Probabilities, and the Hidden Structure of a Betting Market

You look at a horse race on Betfair and see a column of decimal odds. Behind those numbers is a precise geometric structure — one that determines whether the market is fair, whether the bookmaker is taking a cut, and whether a guaranteed profit is sitting there waiting to be collected.

This post builds the vocabulary you need to see that structure. No prior maths required — just a willingness to think carefully about what odds actually mean.

What Are Odds, Really?

Decimal odds as multipliers

On Betfair, odds are displayed in decimal format. If you back a horse at decimal odds of 4.00 with a £10 stake, and it wins, you receive £40 back (your £10 stake plus £30 profit). If it loses, you lose your £10.

The decimal odds tell you the total return per £1 staked:

Higher odds = bigger payout = less likely to win (in the market's opinion).

From odds to probability

Here is the key insight: decimal odds encode a probability. If a horse is priced at 4.00, the market is saying the horse has roughly a 1 in 4 chance of winning. The conversion is:

$$ \text{Implied probability} = \frac{1}{\text{Decimal odds}} $$

We call this the implied probability because it's the probability implied by the price. It might not be the horse's true chance of winning — that's a separate question — but it's the probability embedded in the odds.

Why probabilities must sum to 1

In a three-horse race — say Constitution Hill, State Man, and Energumene — exactly one horse will win. The probabilities of all possible outcomes must add up to 1 (which is 100%):

$$ P(\text{Constitution Hill}) + P(\text{State Man}) + P(\text{Energumene}) = 1 $$

This is not a preference or a convention. It is a logical necessity. If you believe there's a 50% chance of Constitution Hill, a 30% chance of State Man, and a 30% chance of Energumene, your beliefs add up to 110%. You're claiming there's "more than enough probability to go around," which is incoherent — like claiming a pie has more than 100% of its slices.

This is the first constraint that the market must respect.

The Bookmaker's Margin: When Probabilities Don't Add Up

The overround

Look at a traditional bookmaker's odds for a three-horse race:

Sum: $0.556 + 0.333 + 0.200 = 1.089$

The probabilities add up to more than 1. This is called the overround (or "vig" or "juice"). The excess above 1 — here, 0.089 or 8.9% — is the bookmaker's built-in margin. No matter which horse wins, the bookmaker collects slightly more in stakes than they pay out on average.

Think of it this way: the bookmaker is selling probability that doesn't exist. They've created 108.9% worth of probability for an event that can only have 100%. The extra 8.9% is their profit margin.

The exchange difference

On Betfair, you're betting against other people, not against a bookmaker. The exchange takes a small commission on winning bets (typically 2–5%), but the odds themselves are set by supply and demand. Because of competition between bettors, Betfair's implied probabilities often sum to much closer to 1:

Sum: $0.526 + 0.313 + 0.182 = 1.021$

Much tighter — only 2.1% overround. And sometimes, on the back side, the sum drops below 1. When that happens, something remarkable is true: you can back every horse and guarantee a profit. This is called an arbitrage (or "arb" or "sure bet").

Back and Lay: Betfair's Two-Sided Market

What backing and laying mean

On a traditional bookmaker, you can only back a horse — bet that it will win. On Betfair, you can also lay a horse — bet that it will not win. Laying is the opposite of backing: you're taking the bookmaker's role for that selection.

The liability of a lay bet is the amount you'd have to pay if the horse wins:

$$ \text{Liability} = \text{Stake} \times (\text{Odds} - 1) $$

A £10 lay at 4.00 has a liability of £10 × 3 = £30. This is crucial for risk management — a lay bet's risk is not the stake, it's the liability.

The spread

On Betfair, every selection has two prices:

• Back price: the best available odds if you want to bet for the horse (the highest price someone is willing to offer you).
• Lay price: the best available odds if you want to bet against the horse (the lowest price someone is willing to accept).

The lay price is always higher than the back price. The gap between them is called the spread:

Notice the pattern: the spread gets wider for longer-priced horses. This matters because the spread is a cost. If you back Galopin Des Champs at 2.80 and immediately lay at 2.84, you lose a tiny amount. The spread is the price of doing business on an exchange.

The spread is the market's admission charge. Every time you place a bet, you cross the spread. A strategy that generates a 2% edge but pays 3% in spread costs is a losing strategy.

The Order Book: Where Prices Come From

What an order book looks like

Betfair doesn't just show you one back price and one lay price. It shows the order book — a stack of waiting orders at different prices, with amounts attached:

The best back price is 2.78 (the highest someone is willing to offer). The best lay price is 2.84 (the lowest someone is willing to accept). If you want to back this horse, you can get 2.78 for up to £500. If you need more than £500, the next chunk is available at the worse price of 2.76.

This stack of orders is a continuous limit order book (CLOB). The key to understanding it:

• More money at a price = more liquidity (easier to get your bet matched).
• Money at prices far from the best = depth (a large bet won't move the price much).
• Thin books (little money at each price) = illiquid markets where a single bettor can shift the odds significantly.

Why the order book matters

The order book is the raw material of quantitative sports trading. A model that only looks at the best back and lay price is ignoring most of the information available. The shape of the book — how much money sits at each price level, and how fast it arrives and departs — tells you about the market's conviction, the presence of informed money, and the likely cost of executing a bet.

Probability as Geometry: A First Glimpse

The probability triangle

Return to our three-horse race. Every valid probability assignment $(p_1, p_2, p_3)$ satisfies:

$$ p_1 \geq 0, \quad p_2 \geq 0, \quad p_3 \geq 0, \quad p_1 + p_2 + p_3 = 1 $$

This set of all valid assignments is a triangle — the probability simplex. The three corners are absolute certainty:

• $(1, 0, 0)$ — Horse 1 wins for certain
• $(0, 1, 0)$ — Horse 2 wins for certain
• $(0, 0, 1)$ — Horse 3 wins for certain

Every other point inside the triangle represents genuine uncertainty. The centre point $(\frac{1}{3}, \frac{1}{3}, \frac{1}{3})$ means all three horses are equally likely.

When you look at the Betfair prices and compute implied probabilities, you get a point. If that point is inside the triangle, the prices are (at least individually) consistent. If it's outside — because the probabilities are negative, or sum to something other than 1 — the prices contain an inconsistency.

A bookmaker's overround pushes the point outside the triangle (the sum exceeds 1). An exchange arb pushes the point below the triangle on the back side (the sum is less than 1). The distance from the triangle to the point measures the size of the margin or arb.

Why this is just the beginning

For a single win market, the geometry is simple: a triangle (for 3 horses) or a higher-dimensional version of it (for more horses). The consistency check is trivial — add up the implied probabilities.

But real trading involves multiple linked markets: win and place, match odds and correct score, over/under and both teams to score. Each market has its own simplex, but the markets share an underlying reality — the actual match result. The constraints that link them create a more complex shape.

That shape — the marginal polytope — is where the real action is. When two markets individually look fine but are jointly impossible, the combined implied probability vector sits outside the marginal polytope. That's where hidden arbitrages live, and finding them requires understanding the geometry of multi-market consistency.

That's the subject of the next post.

Key Terms Summary

References

1. Boyd, S. & Vandenberghe, L. (2004). Convex Optimization. Cambridge University Press. — Chapter 2 covers convex sets and the probability simplex. Available free online.

2. de Finetti, B. (1931). "Sul significato soggettivo della probabilità." Fundamenta Mathematicae, 17, 298–329. — The founding work on coherent probabilities: if your implied probabilities don't satisfy the simplex constraints, someone can construct a guaranteed profit against you.

3. Shin, H. S. (1991). "Optimal Betting Odds Against Insider Traders." Economic Journal, 101, 1179–1185. — Shows how bookmaker margins (overround) relate to the presence of informed bettors in racing markets.