Turning a price into a chance, and taking the margin back out

§01Decimal odds: the conversion you will use most

Decimal odds state the total returned per unit staked, including the stake. A price of 2.50 returns 2.50 for every 1.00, of which 1.00 is your own money back and 1.50 is profit. The implied chance is the reciprocal: 1 / 2.50 = 0.40, or 40%.

Read the number as an implied chance rather than as a payout and two things become visible. First, longshots are usually priced well below the chance you would assign them, because the margin in a five-way market is spread very unevenly. Second, a move from 2.00 to 1.90 is not "a small move" in probability terms — it is 50.0% to 52.6%, which is a substantial repricing.

Working in probability rather than odds also makes the arithmetic additive. Chances can be compared, averaged and summed; prices cannot. When a market has more than two outcomes, summing the probabilities of every price is the quickest way to see what the operator has taken.

§02Fractional and American prices

Nothing about the underlying idea changes with the notation. Fractional odds of 6/4 mean a profit of 6 for every 4 staked, so the decimal price is 6/4 + 1 = 2.50 and the implied chance is 40%. American prices come in two directions: a positive number is the profit on a 100 stake, so +150 becomes 1 + 150/100 = 2.50; a negative number is the stake required to win 100, so −200 becomes 1 + 100/200 = 1.50, an implied 66.7%.

Convert whatever you are shown into decimal before comparing it with anything else. Mixing formats across the columns of a spreadsheet is the single most common way a personal record ends up describing nothing.

§03The overround: why the two sides add to more than 100

Add the implied chances of every price in a market. On a fair two-way market they would sum to 100%. On a real one they sum to more, and the excess is the margin. Two prices of 1.91 each give 52.4% + 52.4% = 104.7%, a margin of 4.7% of the total. Three outcomes at 3.05, 3.40 and 2.42 give 32.8% + 29.4% + 41.3% = 103.5%.

The margin is the operator's revenue and it is not a hidden charge — it is the reason both sides of any market are shorter than fair. It also varies enormously by market and by operator: a major professional league might be priced at 102%, while a niche market with a low limit can be priced at 112% or worse. The size of the margin is the first thing to check, because it sets the size of the edge you need before anything else is possible.

§04Removing the margin to recover a fair price

To see roughly what the market thinks is fair, strip the margin out. The simplest method is proportional: divide each implied chance by the total. For 1.91 and 1.91, each 52.4% becomes 52.4 / 104.7 = 50.0%, which is the honest answer for a market that was genuinely even. For 1.60 and 2.30, the chances are 62.5% and 43.5% — a total of 106.0% — and the proportional fair chances are 59.0% and 41.0%.

Proportional stripping is an approximation, not a truth: real operators load the margin unevenly, taking far more from longshots than from favourites. A better technique, if you have the data, is to fit the margin across the market and subtract the fitted amount from each outcome. But even the crude version is worth the ten seconds it takes, because the comparison you need is not "price versus my model" but "price versus the de-margined market".

§05A conversion table to work from

Two-way markets, priced at a range of totals, with the margin stripped proportionally. Use it as a sanity check rather than an oracle: the arithmetic is exact, the market it describes is not.

Notice the last two rows. Long prices in the underdog column sit inside much tighter markets than a casual glance suggests, and that is exactly where a wrong estimate is most expensive.