What a price actually says
Every betting price is a probability wearing local clothing. The four common formats differ only in what they take as read - whether the stake is included in the quoted number, and which side of the ratio is fixed at 100 or at 1 - and any of them can be turned into the others without losing information.
Decimal odds quote the total returned per unit staked, stake included. A price of 2.50 returns 2.50 for every 1 risked, of which 1.50 is profit. Because the stake is included, decimal odds are never below 1.00, and the implied probability is simply their reciprocal.
American odds fix one side of the transaction at 100. A negative number is the stake required to win 100: -110 means risk 110 to win 100. A positive number is the profit from a stake of 100: +150 means risk 100 to win 150. The format has a hole in it - nothing between -100 and +100 is a valid price - and 2.00 decimal is the hinge where both branches meet at +100.
Fractional odds, still standard in British and Irish racing, quote profit over stake: 5/2 means two units staked wins five. They exclude the stake, so 5/2 is 3.50 decimal, not 2.50. Hong Kong odds are the same idea in decimal form: profit per unit staked, which is decimal odds minus one.
The conversion, and why the reciprocal is the whole story
Route everything through decimal odds and there is only one formula to remember: p = 1 / d. The reason is worth understanding rather than memorising. A fair bet is one where the expected profit is zero, so if the true win probability is p, then p x (d - 1) = (1 - p) x 1. Rearranged, that is p x d = 1, so p = 1/d. The implied probability is not an opinion about the event; it is the win rate at which the price breaks even.
That identity is why the implied probability and the break-even win rate are the same number, and why this calculator prints them as one figure. If a price implies 52.38%, you need to win 52.38% of such bets to end level. Anything above that is profit; anything below is a slow bleed.
Going back the other way, decimal to American, needs a branch because the American format changes which side is anchored at 100. At d >= 2 the underdog side is anchored, so A = 100(d - 1). Below 2.00 the favourite side is anchored and A = -100/(d - 1). Both expressions give exactly +100 and -100 at d = 2.00, which is the seam.
Fractional odds are the one lossy step, and only cosmetically. Books quote a limited ladder of fractions - 4/5, 10/11, 5/6, evens, 6/5 - so an exact decimal often has no clean fractional equivalent. This calculator finds the closest fraction with a denominator up to 200, so an unusual price may come back as something no book would print. The decimal figure is the exact one.
Worked example: -110 on both sides of a spread
A book prices both teams on a point spread at -110. Take one side with a stake of 100.
- Convert to decimal. The price is negative, so d = 100 / 110 + 1 = 0.909091 + 1 = 1.909091.
- Implied probability. p = 1 / 1.909091 = 0.523810, or 52.3810%.
- Profit and return. Profit = 100 x (1.909091 - 1) = 90.91. Total returned = 100 x 1.909091 = 190.91.
- Fractional. d - 1 = 0.909091, which is 10/11. Hong Kong odds are the same 0.9091.
- Add up both sides. Both teams are priced at 52.3810%, so the book's implied probabilities total 104.7619%. That surplus of 4.7619 percentage points is the overround, also called the vig or juice.
- Strip the margin. Divide each side by the total: 52.3810 / 104.7619 = 50.00% each, which is decimal 2.00 - the fair price with the margin removed.
The conclusion a bettor cares about: you must win 52.38% of -110 bets to break even, not 50%. Over 1,000 bets at 100 a unit, winning exactly half returns 500 wins x 90.91 profit minus 500 losses x 100 = 45,455 - 50,000 = a loss of 4,545. That is the margin doing its work, and it is entirely visible in the conversion.
How to read the implied probability
Compare the implied probability with your own estimate, not with the odds. The price is a threshold: back the bet only when you believe the outcome is more likely than the implied probability. At -110 that threshold is 52.38%; at +150 it is 40.00%; at +1000 it is 9.09%. A long shot is not a bad bet because it is unlikely, it is a bad bet when it is less likely than the price says.
Then check the overround. Add the implied probabilities of every outcome in the market. A two-way market at -110/-110 totals 104.76%, a well-priced two-way market might total 102%, and a 30-runner horse race can total 120% or more. The excess over 100% is the total margin built into the market, and it is the single best comparison between books - far better than eyeballing one price. Removing it properly is what the no-vig fair odds calculator does.
Finally, translate the price into a staking decision rather than a feeling. The expected value calculator combines your probability estimate with the price to give an expected return per unit, and the Kelly criterion calculator turns a positive edge into a stake size. If two books disagree enough that both sides can be backed at a profit, the arbitrage calculator sizes each leg.
Conversion reference for common prices
| American | Decimal | Fractional | Hong Kong | Implied probability |
|---|---|---|---|---|
| -1000 | 1.10 | 1/10 | 0.10 | 90.91% |
| -400 | 1.25 | 1/4 | 0.25 | 80.00% |
| -200 | 1.50 | 1/2 | 0.50 | 66.67% |
| -150 | 1.6667 | 2/3 | 0.6667 | 60.00% |
| -110 | 1.9091 | 10/11 | 0.9091 | 52.38% |
| -105 | 1.9524 | 20/21 | 0.9524 | 51.22% |
| +100 | 2.00 | 1/1 | 1.00 | 50.00% |
| +120 | 2.20 | 6/5 | 1.20 | 45.45% |
| +150 | 2.50 | 3/2 | 1.50 | 40.00% |
| +250 | 3.50 | 5/2 | 2.50 | 28.57% |
| +400 | 5.00 | 4/1 | 4.00 | 20.00% |
| +900 | 10.00 | 9/1 | 9.00 | 10.00% |
| +2000 | 21.00 | 20/1 | 20.00 | 4.76% |
Every row is generated by the same two formulas the calculator uses. Decimal values are rounded for display; the exact figure for -110 is 1.909090..., repeating.
Mistakes that cost money
- Treating implied probability as the true probability. It is the price, and it includes the book's margin. In a two-way market at -110 each side's honest estimate is 50%, not 52.38%.
- Reading fractional odds as including the stake. 5/2 returns 3.50 per unit, not 2.50. The stake comes back on top of the profit.
- Comparing a single price between books instead of the whole market. A book can offer the best price on one side and the worst overall margin. Total the implied probabilities across all outcomes before deciding where to bet.
- Assuming -110 needs a 50% win rate. It needs 52.38%. Across a season of point-spread bets that gap is the difference between a winning and a losing record.
- Averaging odds instead of probabilities. The midpoint of 1.50 and 3.00 is not the midpoint of 66.67% and 33.33%. Convert first, average second, convert back.
- Forgetting that some books quote the stake back separately. Exchange odds are decimal but commission is charged on net winnings, so the effective price is lower than the screen suggests.
Where each format is used
Decimal odds dominate Europe, Australia and betting exchanges, and are the default in almost all quantitative work because they multiply cleanly across the legs of a parlay. American odds are standard in North American sportsbooks. Fractional odds survive in British and Irish horse and greyhound racing and in the traditional UK high street. Hong Kong odds appear in Asian handicap markets, where Malay and Indonesian formats also circulate; those two behave like Hong Kong odds with sign conventions for favourites and underdogs, and are not covered here.
What to do after converting
Conversion is a first step, not a strategy. Once every price is on the same scale, three questions follow. First, what is the market's margin, and is it small enough that a genuine edge can survive it? Second, does your own probability estimate beat the implied probability by enough to matter, given that your estimate is itself uncertain? Third, how much should you stake given a bankroll you would like to keep?
The arithmetic of multi-leg bets follows directly from decimal odds, which is another reason to work in them. A parlay's decimal price is the product of its legs: three legs at 1.909091 make 1.9090913 = 6.9579, an implied probability of 14.37%, and the margin compounds with every leg added. The parlay payout calculator handles that arithmetic and shows how quickly the accumulated vig eats the extra return.
It is also worth knowing what implied probability is not. It is not calibrated: a market that prices a hundred events at 20% will not necessarily see twenty of them land, particularly at long prices, where the well-documented favourite-longshot bias means long shots are systematically overpriced relative to their frequency. And it is not stable: closing lines move as money arrives, and the closing price is generally the most accurate probability estimate available on an event. Comparing the price you took with the closing price is the cleanest self-assessment a bettor has.
Key terms
- Implied probability
- The reciprocal of decimal odds, expressed as a percentage. It is the win rate at which the bet breaks even, and it includes whatever margin the book has built in.
- Overround (vig, juice)
- The amount by which the implied probabilities of all outcomes in a market exceed 100%. A two-way market at -110 on both sides has an overround of 4.76 percentage points.
- Break-even win rate
- The proportion of bets at a given price that must win for the profits to exactly cover the losses. Numerically identical to the implied probability.
- Closing line
- The final price before an event starts. It reflects all the money bet and is the standard benchmark against which a bettor's price is judged.