Decimal, Fractional and American Odds, Converted

Prediction markets quote in cents or dollars, which is already a probability. Betting markets quote in decimal, fractional or American odds. If you move between them, here are the conversions and the one habit worth adopting.

The four formats

Probability (%) — what a prediction market quotes. 62 cents means 62%.

Decimal odds — total return per unit staked, including the stake. 1.61 means a $1 stake returns $1.61 in total.

Fractional odds — profit relative to stake. 8/13 means a $13 stake wins $8 profit.

American odds — for favourites, the stake needed to win $100 (negative). For underdogs, the profit on a $100 stake (positive). −162 means stake $162 to win $100. +110 means stake $100 to win $110.

Converting to probability

This is the only conversion you actually need, because probability is the unit everything should be compared in.

From decimal:     p = 1 ÷ decimal
From fractional:  p = denominator ÷ (numerator + denominator)
From American −:  p = |odds| ÷ (|odds| + 100)
From cents:       p = cents ÷ 100

Worked, all pointing at the same place:

decimal 1.61    → 1 ÷ 1.61    = 0.621
fractional 8/13 → 13 ÷ 21     = 0.619
American −162   → 162 ÷ 262   = 0.618
62 cents        → 62 ÷ 100    = 0.620

Converting from probability

To decimal:       decimal = 1 ÷ p
To American (p ≥ 0.5): odds = −(100 × p) ÷ (1 − p)
To American (p < 0.5): odds = (100 × (1 − p)) ÷ p
To cents:         cents = p × 100

A 62% probability is decimal 1.61, American −163, and 62 cents.

The reference points worth memorising

Probability	Decimal	American
90%	1.11	−900
80%	1.25	−400
75%	1.33	−300
66.7%	1.50	−200
50%	2.00	±100
33.3%	3.00	+200
25%	4.00	+300
20%	5.00	+400
10%	10.00	+900

Five or six of these in your head is enough to sanity-check almost anything without a calculator.

The vig only shows up in probability

Here is why converting to probability matters rather than being a preference.

A sportsbook offers both sides of an event at −110. Convert each: 110 ÷ 210 = 52.4%. Two sides, both at 52.4%, sum to 104.8%. That 4.8% over 100 is the house margin, and it is completely invisible in the odds format — "−110 both ways" looks symmetric and fair.

Now do the same on an exchange. Yes offered at 52, No offered at 49. Sum: 101%. The round-trip cost is about 1%.

Both numbers are sitting right there once you convert, and neither is visible before you do. This is the whole argument for the habit: convert everything to probability, add up the outcomes, and look at the excess over 100%. That excess is what you are paying, whoever you are paying it to.

Prediction markets versus sportsbooks works through what that difference means structurally.

A note on where the excess goes

On an exchange, the excess over 100% is the spread, and it goes to whoever provided liquidity, plus the venue's fee. On a book, it goes to the book. In both cases it comes out of participants in aggregate, and it is the first thing any supposed edge has to clear. How fees eat a theoretical edge does the arithmetic on what survives.

Multi-outcome markets

The same check works with more than two outcomes: convert every outcome to a probability and add them all. A field of eight candidates summing to 108% means eight spreads, and buying the whole field costs you eight cents on the dollar to guarantee a dollar.

If a field sums to less than 100%, either a quote is stale or the field is not exhaustive — there is an "anybody else" outcome that is not listed. When the outcomes do not add to 100% covers both.

Next

The favourite-longshot bias is the most durable systematic distortion in these numbers, and it shows up in every format equally.