Mathwords logoMathwords

Odds Against — Definition, Formula & Examples

Odds Against

Odds m:n (read aloud "m to n") against an event mean we expect the event will not to occur m times for every n times it does occur.

When the odds against an event are m:n,m:n\text{,} the probability the event will occur is nm+n.\displaystyle \frac{n}{m+n}\text{.}

Example: The odds against rolling a 3 on a fair six-sided die are 5:1 since this event has a 1/6 probability. We expect five rolls of numbers other than 3 for every one roll that is a 3.

See also

Odds in favor, odds in gambling

Key Formula

Odds against=P(not E)P(E)=1−P(E)P(E)\text{Odds against} = \frac{P(\text{not } E)}{P(E)} = \frac{1 - P(E)}{P(E)}
Where:
  • P(E)P(E) = The probability that the event E occurs
  • P(not E)P(\text{not } E) = The probability that the event E does not occur, equal to 1 − P(E)

Worked Example

Problem: A standard die is rolled once. What are the odds against rolling a 5?
Step 1: Count the favorable outcomes. Only one face shows a 5, so there is 1 favorable outcome.
Favorable outcomes=1\text{Favorable outcomes} = 1
Step 2: Count the unfavorable outcomes. The remaining faces (1, 2, 3, 4, 6) give 5 unfavorable outcomes.
Unfavorable outcomes=5\text{Unfavorable outcomes} = 5
Step 3: Form the ratio of unfavorable to favorable outcomes.
Odds against rolling a 5=5:1\text{Odds against rolling a 5} = 5 : 1
Answer: The odds against rolling a 5 are 5 to 1. This means for every 1 time you expect to roll a 5, you expect 5 times that you will not.

Another Example

Problem: A bag contains 3 red marbles and 7 blue marbles. You draw one marble at random. What are the odds against drawing a red marble?
Step 1: Identify favorable outcomes (drawing red) and unfavorable outcomes (drawing blue).
Favorable=3,Unfavorable=7\text{Favorable} = 3, \quad \text{Unfavorable} = 7
Step 2: Write the ratio of unfavorable to favorable.
Odds against red=7:3\text{Odds against red} = 7 : 3
Answer: The odds against drawing a red marble are 7 to 3.

Frequently Asked Questions

How do you convert odds against to probability?
If the odds against an event are m : n, the probability that the event occurs is n / (m + n). For example, if the odds against are 5 : 1, the probability of the event is 1 / (5 + 1) = 1/6.
What is the difference between odds against and odds in favor?
Odds against place unfavorable outcomes first (unfavorable : favorable), while odds in favor place favorable outcomes first (favorable : unfavorable). If the odds against are 5 : 1, the odds in favor are simply 1 : 5. They convey the same information but from opposite perspectives.

Odds Against vs. Odds in Favor

Why It Matters

Odds against appear frequently in gambling, sports betting, and everyday risk assessment. When a horse is listed at 9 to 1, those are odds against — telling you the horse is expected to lose 9 times for every 1 time it wins. Understanding this ratio helps you evaluate risk and calculate potential payouts from stated odds.

Common Mistakes

Mistake: Confusing odds against with probability. For example, saying the odds against rolling a 5 are "1 in 6."
Correction: Odds are a ratio of two counts (unfavorable to favorable), not a single fraction out of the total. The probability of rolling a 5 is 1/6, but the odds against are 5 : 1.
Mistake: Reversing the ratio — putting favorable outcomes first when stating odds against.
Correction: For odds against, unfavorable outcomes always come first. If you accidentally write 1 : 5 instead of 5 : 1, you have stated the odds in favor, not against.

Related Terms

  • Event — The outcome whose odds are being described
  • Odds in Favor — The reverse ratio: favorable to unfavorable
  • Odds in Gambling — How odds set payouts in betting
  • Probability — Fraction form of likelihood, convertible to odds
  • Ratio — The mathematical form used to express odds
  • Sample Space — Set of all outcomes from which odds are calculated