Odds and probability are the same fact in two dialects
A probability splits the world into success against everything. Odds split it into success against failure. A probability of 0.75 says three quarters of outcomes succeed; odds of 3:1 say the same thing as a head-to-head: three wins for every one loss. Bookmakers quote odds because stakes and payouts compare naturally in that form. Statisticians quote probability because it adds and averages cleanly. Converting between them is two one-line formulas, and getting fluent in both directions stops a family of expensive misunderstandings.
A raffle seller has won 18 of her last 24 weekly draws at a game with fixed odds. Her win rate is 18 ÷ 24 = 0.75, which as odds is 18:6, reduced to 3:1 in favour. Run the formula: 0.75 ÷ 0.25 = 3. Backwards: 3 ÷ (3 + 1) = 0.75. The two dialects meet exactly, as they must.
Reading odds against, fractional and decimal
Odds against flip the ratio: the same event is 1:3 against. British fractional odds like 5/1 are odds against written as a fraction, profit per unit staked. Decimal odds (2.50, 4.00) include the returned stake: decimal = 1 ÷ p, so a fair 3:1 shot prices at 4.00. A bookmaker's prices always imply probabilities summing to more than 1; the excess is the margin, and spotting it starts with converting every price back to a probability with the tool above.
Treating "3 to 1" as a probability of 1/3. It is 3/(3+1) = 0.75 in favour, or 0.25 if the quote was odds against. The words "in favour" or "against" carry the whole meaning; when a quote does not say, ask. Sports coverage is loose about this, and the two readings differ by a factor of nine here.
Frequently asked questions
How do I convert odds of 5:1 to a probability?
If the 5:1 is odds against, the probability is 1/(5+1) = 0.167. If it is odds in favour, it is 5/(5+1) = 0.833. Check which direction the quote runs before converting.
What are even odds?
1:1, a probability of exactly 0.5. A fair coin flip is the standard example.
Why do bookmakers' implied probabilities add to more than 100%?
The margin. Each price is shaded slightly against the bettor, so the implied probabilities sum to something like 105%. The extra 5% is the bookmaker's expected cut.