“The Language of Odds, Probability, and Expected Value”

A recent academic study presented at the 19th International Conference on Gambling & Risk Taking explores how probability and expectation work in real sports betting markets. The paper, titled "Beat the Sportsbook: The Theory and Practice of Successful Sports Betting", was created by Robert Scott, a professor at Monmouth University, and Mikhail Sher, an associate professor at Monmouth University.

In sports betting, nothing meaningful starts with the odds themselves. Odds are just a way of turning probability into something that can be traded.

Behind every price is an implied likelihood, and behind that likelihood is a set of assumptions about how often something should happen.

Decimal odds can be converted into probability by taking the reciprocal. A price of 2.00 implies a 50% chance. A price of 1.50 implies a 66.7% chance.

But these are not neutral probabilities. They are bookmaker probabilities, shaped by margin and market behaviour. The bookmaker is not trying to describe the truth; they are trying to balance risk.

The bettor's task is always the same: compare the implied probability with a more realistic estimate. If there is a gap in your favour, there may be value. If not, there is nothing to exploit.

The Monmouth University research looks closely at how the gap between implied probability and more realistic estimates plays out in actual betting markets. It uses this idea as a base for exploring whether sports bettors can realistically discover and gain an edge against the sportsbook over time, or whether things like the vig make this difficult.

Talking about how significant the vig is as a structural barrier for sports bettors, Robert Scott told us, "It's massive. You have to pick winners over 52.4% of the time to overcome the vig-and these are moneyline bets at -110. If we're talking about parlays, then it's even worse. Most people never come close to beating the vig because they place bets with high cost margins with far too little chance of winning. This isn't to say parlays are bad-they aren't, but you have to play them smart and bet them as the long-shorts they are."

Expectation: the long-run average hidden in uncertainty

To do that properly, you need to understand expectation.

Expected value is simply the long-run average outcome of a random process, weighted by probability. It tells you what would happen if you could repeat the same bet an infinite number of times under identical conditions.

The idea dates back to the early development of probability theory in the 17th century. When Blaise Pascal was asked to solve the "problem of points" in a popular game of chance, he and his colleague Pierre de Fermat formalised the idea that uncertain outcomes could be valued mathematically rather than guessed. That work effectively created the foundation for expected value.

At its core, expectation is just a weighted average: each outcome multiplied by its probability, then summed. For example, with a fair die:

· (1 + 2 + 3 + 4 + 5 + 6) ÷ 6 = 3.5

The key point is that 3.5 is not a possible roll. It is the centre of the distribution, not a prediction of a single event. Expectation describes the long-run tendency of a process, not its next outcome.

This idea of expectation is also a key focus in the Monmouth University research, particularly when looking at how sports bettors try to apply it to real markets. The study talks about Positive Expected Value (+EV) betting, which is a data-driven strategy that aims to recognise wagers where the probability of winning is higher than what the sportsbook's odds suggest, and explores whether +EV betting can realistically overcome the sportsbook's margin over time.

Expected value in practice

The same idea applies to any random system, including sports.

In football analytics, expected goals (xG) is a direct application of expectation. Each shot is assigned a probability of becoming a goal based on historical data: location, angle, defensive pressure, and so on.

If similar shots are scored 4 times out of 10, the expected value of a shot in that category is 0.4 goals. A match becomes a collection of these small probabilistic events, and a team's total xG is the sum of all individual shot values.

This matters because actual goals are noisy. A team can win comfortably while creating fewer chances, or they can lose while dominating expected output. Over small samples of matches, randomness dominates. xG attempts to filter that noise and reveal the underlying signal.

Expected value in betting: finding the edge

In betting, expected value becomes a financial measure of whether a wager is worthwhile in the long run. The basic structure is:

· Expected value = (probability of winning × profit if you win) − (probability of losing × loss if you lose)

Take a simple example. Suppose you believe a tennis player has a 60% chance of winning. The bookmaker offers odds of 1.72. If your probability is correct, the fair odds are closer to 1.67. The difference creates a small edge. For a £1 stake:

· (0.6 × £0.72) − (0.4 × £1) = £0.032

This £0.032 is not a guaranteed profit. It is the average outcome over many repeated bets. In reality, any single bet will either win or lose. But over a large enough sample, the randomness evens out, and the expectation becomes visible.

There is also a shortcut: divide bookmaker odds by "true" odds. If the result is above 1, there is positive expectation. If it is below 1, there is not. In this example, 1.72/1.67 = 1.032. This is your expected return, or £1.032 from a £1 stake.

Clearly, expected profit is expected return minus 1. The difficulty is obvious: "true" odds are never known precisely. They are always estimates, built on models that are themselves imperfect.

Accumulators and compounding expectation

Because probability multiplies across independent events, expected value also compounds. If two bets each have an expected return of 1.05, then a double has:

· 1.05 × 1.05 = 1.1025

This is why accumulators are attractive. Small edges appear to grow quickly when combined. A treble or fourfold can look like a powerful amplifier of value. But there are two important consequences.

First, the probability of success collapses rapidly. Even with fair 50/50 bets, a four-fold wins only 6.25% of the time. The higher return comes with much higher variance.

Second, any error in your probability estimates is also compounded. If your "true" odds are wrong, even slightly, the multiplication works against you just as quickly. What looks like compounding profit can become compounding loss.

This is why bookmakers encourage accumulators: they increase turnover, variance, and margin exposure simultaneously, knowing that most of their customers don't really hold an edge even if the customers themselves believe that they do.

Why expectation is not enough

Expected value tells you whether a strategy is profitable in theory. But it does not describe the experience of following that strategy.

Two bettors can have identical expected value and yet live in completely different realities. One might experience smooth growth. The other might suffer long losing streaks before recovering. Both outcomes are consistent with the same expectation.

This is because expectation is only the centre of the distribution. It does not describe spread, volatility, or sequence.

In betting, that distinction matters. Many bettors fail not because their ideas are wrong in expectation. They fail because they cannot tolerate the variance around it.

From expectation to simulation

This is also something that the Monmouth University research highlights in a wider sense. Even when sports bettors understand expectation and value, real betting outcomes don't always follow a predictable path. Results can be shaped by random events and market conditions, which means short-term performance can look different from what the maths suggests.

When asked about where the line is between short-term success and genuine long-term edge, Robert Scott told us, "In sports betting, there is only the short-term. Everything is based on a bet-by-bet basis. If you're betting good odds at good prices, you will win in the long run, but you never think of the long run."

To understand that variance properly, you need more than a single number. You need to visualise the full range of possible outcomes. That is where simulation becomes useful.

Instead of calculating only the average result, you can simulate thousands of versions of the same betting process. Each simulation introduces randomness: wins, losses, streaks, and variation in outcomes.

Across those simulations, you begin to see patterns that expectation alone hides: drawdowns, recovery periods, losing runs, and the probability of ruin.

In other words, expected value tells you what should happen on average. To visualise the range of possible outcomes, we should turn to a tool called the Monte Carlo simulation to reveal what can actually happen when randomness is allowed to unfold repeatedly.

And in sports betting, that difference is often the most important part of all. In my next article, we will take a closer look at the Monte Carlo simulation.