On this page
A question says "the odds in favour of an event are 3:5" and asks for its probability. Many students write 3/5, and lose the mark. Odds and probability measure the same thing on different scales, and mixing them up is one of the easiest errors in the chapter. This post explains the difference, gives conversion rules in both directions, works through combined events, and shows where odds still turn up outside the exam hall.
Is "odds" in the NCERT books and the syllabus?
Not by name. The word does not appear in the current NCERT probability chapters for Class 11 (Chapter 14, 2026-27 reprint) or Class 12 (Chapter 13, 2026-27 reprint), and the JEE (Main) 2026 syllabus lists "probability of an event, addition and multiplication theorems of probability, Baye's theorem" without mentioning odds. Odds is simply another way of stating a probability, so questions that use it are testing the same syllabus content. You will meet it in older textbooks, reference books and practice papers, so it is worth five minutes. Check the current syllabus for your exam.
What is the difference between odds and probability?
Probability compares favourable outcomes with all outcomes. Odds compare favourable outcomes with unfavourable ones.
A BMJ statistics note by Bland and Altman uses a die: the odds that a single throw produces a six are 1 to 5, because one face is a six and five are not. The probability of a six is 1/6. The note defines the odds as the ratio of the probability that an event occurs to the probability that it does not.
So if p is the probability:
Odds in favour = p : (1 − p)
Odds against = (1 − p) : p
How do you convert odds to probability?
If the odds in favour are a:b, think of a + b equally likely outcomes, a of them favourable.
Odds in favour a:b → P(event) = a/(a + b).
Odds against a:b → P(event) = b/(a + b).
Example 1. Odds in favour of A are 3:5. Then P(A) = 3/(3 + 5) = 3/8, and P(not A) = 5/8.
Example 2. Odds against B are 7:2. Then P(B) = 2/(7 + 2) = 2/9.
Example 3. P(C) = 0.2 = 1/5. Then P(not C) = 4/5 and the odds in favour are 1/5 : 4/5 = 1:4. The odds against are 4:1.
Example 4. Two fair dice are thrown. Six of the 36 outcomes give a sum of 7, so P = 6/36 = 1/6. Odds in favour are 1:5 and odds against are 5:1.
How do you combine odds for two events?
Never multiply or add odds directly. Convert to probabilities, combine using the rules you know, then convert back.
Worked example. The odds in favour of A are 2:3 and the odds in favour of B are 3:4. A and B are independent. Find the odds in favour of (i) both happening and (ii) at least one happening.
Convert: P(A) = 2/5 and P(B) = 3/7.
Both: P(A and B) = (2/5)(3/7) = 6/35. The other 29/35 is "not both", so the odds in favour are 6:29.
At least one: P(neither) = (3/5)(4/7) = 12/35, so P(at least one) = 1 − 12/35 = 23/35. The odds in favour are 23:12.
Check step 3 with the addition theorem: P(A or B) = 2/5 + 3/7 − 6/35 = 14/35 + 15/35 − 6/35 = 23/35. It matches. For the independence rule itself, see our guide to independent events in probability.
When do odds and probability nearly agree?
Odds = p/(1 − p). When p is small, 1 − p is close to 1, so the odds are close to the probability. For p = 0.01, the odds are 1/99 ≈ 0.0101. For p = 0.5 the odds are 1, double the probability, and for p = 0.9 they are 9.
This is why odds matter beyond exams. Medical studies often report odds ratios. A paper in the BMJ by Davies, Crombie and Tavakoli points out that odds ratios are often read as if they were relative risks, and that doing so always overstates the effect, more so when the outcome is common. The maths is the same as above: for common events, odds and probability are far apart.
Which mistakes cost marks?
Writing a/b for odds a:b. The probability is a/(a + b).
Mixing "in favour" and "against". Read which side comes first. Odds against 7:2 means seven unfavourable to two favourable.
Multiplying odds for independent events. Multiply probabilities, then convert.
Forgetting to simplify. Odds of 6:36 and 1:6 are the same; reduce before you answer.
Treating odds as a fraction of a whole. Odds of 1:1 means a probability of 1/2, not 1.
For more slips like these across the chapter, read our list of common probability mistakes. If you want to check a conversion step by step, the Fermi Problem Library has worked problems with a step-by-step solver that explains the reasoning rather than printing an answer.
A practice routine
Convert ten probabilities to odds in favour and against, and ten odds back to probabilities, in under five minutes.
A card is drawn from a pack of 52. Find the odds in favour of a king. Check: 4:48, which is 1:12.
Redo the worked example with A at 1:2 and B at 1:3. Check: odds in favour of at least one are 1:1.
Write one sentence explaining why odds and probability agree for rare events but not for common ones.
Tonight, redo example 1 and the worked example from a blank page, writing the a + b step every time.
