Betting Odds and Implied Probability Cheat Sheet by BetBetter

Decimal 2.50 = stake back x 2.50 (a $10 bet returns $25, profit $15)
Fractional 3/2 = profit 3 for every 2 staked (same as 2.50)
American +150 = profit $150 on a $100 bet (same as 2.50)
American -200 = stake $200 to win $100 (decimal 1.50)

Convert:
Decimal = 1 + fraction (3/2 -> 2.50)
Decimal from +A = 1 + A/100
Decimal from -A = 1 + 100/A
+A from decimal d >= 2: (d - 1) x 100
-A from decimal d < 2: 100 / (d - 1)

Free converter: betbet­ter.wo­rld­/to­ols­/od­ds-­con­verter

Calibr­ation = do events priced at X% happen about X% of the time?

1. Remove the margin from each closing price
2. Group outcomes into bands (50-60%, 60-70% ...)
3. Compare the average priced chance with the share that won
4. Judge gaps against the standard error: sqrt(p x (1-p) / n)

What open data shows: across 18,238 settled two-ou­tcome markets (Nov 2024 - Jul 2026, five sports), outcomes landed within about 2 points of the margin­-free price in every band. Small gaps like that are often inside sampling error.

Full dataset (36,476 rows, CC BY 4.0): DOI 10.528­1/z­eno­do.2­17­64674
Study: Do betting odds tell the truth?

Decimal d: chance = 1 / d
2.50 -> 1 / 2.50 = 40%
1.50 -> 66.7%

Fractional a/b: chance = b / (a + b)
3/2 -> 2 / 5 = 40%

American +A: chance = 100 / (A + 100)
+150 -> 100 / 250 = 40%

American -A: chance = A / (A + 100)
-200 -> 200 / 300 = 66.7%

Short-cut: an even-money price (2.00, 1/1, +100) means 50%.

Implied probab­ility: the chance a price suggests (1 / decimal odds)
Margin / vig / overround: how far the implied chances add up above 100%
Fair (no-vig) odds: odds after the margin is removed
Closing price: the last price before the event starts
Favour­ite­-lo­ngshot bias: favourites and longshots being priced too short or too long on average
Brier score: average squared error of a forecast chance (a coin flip scores 0.25)
Calibr­ation: priced chances match how often things happen

Both sides of a two-way market add up to more than 100%. The extra is the bookma­ker's margin.

Example: Team A 1.80, Team B 2.10
Implied: 1/1.80 = 55.6%, 1/2.10 = 47.6%
Total = 103.2% -> margin about 3.2%

Fair chance (simple method): divide each by the total
A = 55.6 / 103.2 = 53.9%
B = 47.6 / 103.2 = 46.1%

Fair odds = 1 / fair chance
A = 1.86, B = 2.17

Other methods (Shin, power, odds ratio) spread the margin differ­ently between favourite and longshot.