Expected value from first principles
Imagine you could place the same bet ten thousand times. Some fraction wins, the rest lose, and the average outcome across all of them is the expected value. It is not what happens on any single bet — a positive-EV bet still loses regularly — it is the long-run centre of gravity that decides whether repeating the bet builds a bankroll or drains it.
Written out, EV weighs each outcome by how likely it is and how much it pays.
# EV of a bet, per 1 unit staked # p = your probability the bet wins # D = decimal price # (D − 1) = profit per unit if it wins EV = p × (D − 1) − (1 − p) × 1 # win branch pays profit (D − 1); lose branch forfeits the 1 unit stake
If EV is positive the bet is, on average, priced in your favour. If it is negative it is priced against you. At the posted price almost every bet is negative EV — that is the book's margin doing its job. Finding the rare positive-EV bet is the entire game, and it starts with a probability you trust more than the market's.
A worked example
Suppose a two-way market offers decimal 2.10 on the side you like, and your own analysis puts the probability of that outcome at 52 percent. Everything below is illustrative — the point is the arithmetic, not the specific numbers.
# D = 2.10, your p = 0.52 profit_if_win = D − 1 = 1.10 EV = 0.52 × 1.10 − 0.48 × 1 = 0.572 − 0.48 = +0.092 # about +9.2% expected return per unit staked
Positive — but read the condition carefully. That +9.2 percent exists only if 0.52 is a better estimate of the true probability than the market's. Change your estimate to 0.47 and the same price flips to negative EV. The edge lives entirely in the accuracy of p.
Value is the gap, not the result
The most common mistake is treating a likely winner as a good bet. Value has nothing to do with whether an outcome is probable and everything to do with whether it is underpriced. Formally, a bet has value when your fair probability exceeds the price's fair probability.
| Case | Decimal | Market fair p | Your p | Value? |
|---|---|---|---|---|
| Short favourite | 1.40 | 71.4% | 70% | No — you're worse |
| Even pick | 2.00 | 50.0% | 54% | Yes — you're better |
| Long shot | 5.00 | 20.0% | 23% | Yes — you're better |
The favourite is the most likely winner and the worst bet of the three, because your estimate sits below its fair probability. The long shot loses most of the time yet carries value, because you price it higher than the market does. Likelihood and value are different axes.
Anchoring to a fair line
The weak link is always your probability estimate, so disciplined bettors do not pull it from thin air. One rigorous baseline is the fair line from a tight, low-margin market: take the raw implied probability, remove the overround by normalising, and treat that margin-free figure as the market's honest estimate. Your edge is then the difference between your number and that fair anchor — not the raw price.
# 1. raw implied prob from decimal: p_raw = 1 / D # 2. remove margin (normalise): p_fair = p_raw / book_pct # 3. your edge: edge = p_you − p_fair # positive edge is the precondition for a positive-EV bet at that price
The margin-removal step is covered in full in the implied probability and overround guide, and the fair-line calculator automates it. Beating the closing number of a sharp market over time — closing line value — is the clearest external evidence that your estimates carry a real edge rather than noise.
From EV to a decision
Expected value answers whether to bet. It does not answer how much — that is a separate question about bankroll and the reliability of your edge. The honest workflow is a short chain: estimate a probability you can defend, remove the market's margin to get a fair anchor, confirm your number beats it, compute EV to confirm the price is worth taking, and only then size the stake. Skip the fair-line step and the EV number is measured against the book's margin instead of its true view, which quietly overstates every edge.