Probability and Odds: What the Numbers Mean

Probability measures the chance of an event happening. Odds compare that chance with the chance it does not happen. A 60% probability therefore means odds of 3 to 2 in favor.

Prediction markets introduce another number: the price of a YES share. For a binary contract that pays $1 if YES wins and $0 otherwise, a 60-cent price translates into a 60% implied probability before costs. That is a price-based forecast—not proof that the event’s true chance is 60%.

How to convert probability and odds

Write probability as a decimal, p: 60% becomes 0.60. The probability of the event not happening is 1 − p.

The conversion is:

Odds in favor = p / (1 − p)

For 60%, that gives 0.60 / 0.40 = 1.5, or 3:2 in favor. UCLA’s statistical guidance defines odds this way: the probability of success divided by the probability of failure. At 100% probability, the denominator is zero, so there is no finite odds ratio.

To reverse the calculation:

• Odds a:b in favor → probability = a / (a + b)
• Odds a:b against → probability = b / (a + b)

So 4:1 in favor means 4/5 = 80%, while 4:1 against means 1/5 = 20%. Always establish which direction the odds describe.

Statistical odds are not payout odds

Betting formats create an easy source of confusion. Statistical odds compare occurrence with non-occurrence; fractional betting odds describe profit relative to the stake.

An event with a 20% probability has statistical odds of 1:4 in favor. Its equivalent fair fractional payout is 4/1: four units of profit per unit staked if it wins.

The following are mathematical equivalents, assuming no fees or bookmaker margin—not live market quotes:

Probability	Statistical odds in favor	Fractional payout odds	Decimal payout odds	American odds
20%	1:4	4/1	5.00	+400
50%	1:1	1/1	2.00	+100
60%	3:2	2/3	≈1.667	−150
80%	4:1	1/4	1.25	−400

To convert a quoted payout into implied probability:

• Decimal odds D: p = 1 / D. Decimal odds include the returned stake.
• Fractional odds a/b: p = b / (a + b).
• American odds +A: p = 100 / (A + 100). For negative odds −A, use the magnitude: p = A / (A + 100).

These are the conversion formulas explained by Smarkets. They translate a price; they do not establish forecast accuracy or remove bookmaker margin. For example, two mutually exclusive, exhaustive outcomes each quoted at decimal odds of 1.90 imply about 52.6% each—a total of 105.3%, not 100%.

Why a 60-cent YES share implies 60%

Polymarket’s explanation of its contracts specifies that winning outcome shares pay $1 USDC upon resolution, while losing shares become worthless.

For that payoff structure, suppose a hypothetical YES share costs $0.60 and is held to resolution. Ignoring fees and the time value of money:

• If YES wins, the net profit is $0.40.
• If NO wins, the loss is $0.60.

At a probability p, expected net profit is:

p × $0.40 − (1 − p) × $0.60 = p × $1 − $0.60

It breaks even at p = 0.60. This is the arithmetic behind the price-to-probability translation. See how prediction-market prices become probabilities for the broader mechanism and limitations.

But distinguish the displayed number from the executable price. Polymarket’s guidance, dated March 13, 2026, says its displayed probability normally uses the bid–ask midpoint, switching to the last traded price when the spread exceeds $0.10. A midpoint is not necessarily a price at which you can buy.

Costs also change the arithmetic. If the hypothetical 60-cent purchase carries a 2-cent upfront fee and no other charges, the total cost is 62 cents, so its hold-to-resolution break-even probability is 62%, not 60%, still ignoring time value.

Conversion is not calibration

An 80% forecast still allows a 20% chance of failure. One losing favorite therefore does not, by itself, show that the probability was wrong.

Calibration asks whether events assigned similar probabilities happen at approximately those rates over a sufficiently large sample. For example, among forecasts near 80%, roughly 80% should resolve YES. This is the standard calibration principle, not a claim that any particular market meets it.

For a prediction-market evaluation, record the contract’s exact resolution rule, forecast timestamp, price measure and eventual outcome. Compare similar forecast horizons rather than mixing early prices with near-resolution prices. Then examine observed YES frequencies by probability range; Brier score and calibration answer related but different questions.

The distinction is practical: conversion tells you what a quoted number implies; calibration tests whether those implications match the record.