The formulas, side by side
Probability = favorable outcomes ÷ total outcomes — a single number between 0 and 1 (or 0% and 100%), since it's a fraction of everything that could happen. Odds = unfavorable outcomes : favorable outcomes — a ratio comparing the two outcome groups directly to each other, not to the whole. This is the core structural difference: probability's denominator is the total; odds' "denominator" is just the other outcome group.
A worked example — rolling a die
Rolling a 5 or 6 on a fair six-sided die: 2 favorable outcomes out of 6 total. Probability = 2 ÷ 6 ≈ 33.33%. Odds = the 4 unfavorable outcomes to the 2 favorable ones, reduced to 2:1 against — meaning for every 1 favorable outcome, there are 2 unfavorable ones. Both describe the exact same underlying event, but they're genuinely different numbers (33.33% vs a 2:1 ratio) answering slightly different questions.
A worked example — drawing a card
Drawing an ace from a standard 52-card deck: 4 favorable outcomes out of 52 total. Probability ≈ 7.69%. Odds = the 48 unfavorable outcomes to the 4 favorable ones, reduced to 12:1 against. Notice odds numbers can look much more dramatic than the probability percentage for the same rare event — "12:1 against" and "7.69% chance" describe the identical likelihood, but the odds framing emphasizes the gap between outcome groups rather than the share of the whole.
Why gambling and betting contexts favor odds
Betting odds are typically expressed as a ratio because they map directly onto payout structure — "2:1 against" naturally suggests a bet that pays 2 units for every 1 unit risked to break even on the underlying probability. Probability as a percentage doesn't translate as directly into a payout ratio, which is part of why betting markets, horse racing, and casino games traditionally quote odds rather than probability, even though the two carry the exact same information about likelihood.
How to convert between them
From odds "a:b against" (a unfavorable, b favorable) to probability: b ÷ (a + b). From probability p to odds: (1 − p) : p, then simplified to smallest whole numbers. The two representations always carry identical information — knowing one lets you derive the other exactly, so a claim like "a 2:1-against event has a 40% probability" is simply wrong (2:1 against converts to 1 ÷ 3 ≈ 33.33%, not 40%) and worth catching whenever odds and probability figures are quoted together for the same event.