Odds

Definition:
If there are a ways to "win" and b ways to "lose",
then the odds in favor are a:b and the odds against are b:a.
If an event E has a favorable outcomes and b unfavorable outcomes,
then the odds in favor of the event E occurring are a:b and the odds against E occurring are b:a.

Ex. There are 3 ways to win and 8 ways to lose,
thus the odds in favor of winning are 3:8
and the odds against winning are 8:3.
Ex. There are 5 red balls and 12 non-red balls to randomly choose one from.
So the odds in favor of choosing a red ball are 5:12
and the odds against choosing a red ball are 12:5.

Odds to probability:

Ex. The probability of winning is 3 / 3+8, or 3/11 ≈ .2727 and
the probability of losing is 8 / 3+8, or 8/11 ≈ .7272
Ex. The probability of choosing a red ball is 5 / 5+12, or 5/17 ≈ .2941 and
the probablity of not choosing a red ball is 12 / 5+12, or 12/17 ≈ .7059

If the odds in favor of event E are a:b, then the probability that the event occurs is P(E) = a / a+b

Probability to odds:

Ex. If the probability of winning is 3/11, then the odds in favor of winning are 3:11-3, or 3:8 and the odds against winning are 11-3:3, or 8:3.
Ex. If the probability of winning is 5/17, then the odds in favor of winning are 5:17-5, or 5:12 and the odds against winning are 17-5:5, or 12:5.

If the probability of event E is P(E), then the odds in favor of it are P(E) / 1-P(E) and the odds against it are 1-P(E) / P(E).

Ex. If the probability of an event is .294, then the odds in favor of it are .294/.706 = 294:706, and the odds against it are .706/.294 = 706/294.

The "actual odds against" is the ratio of the probability of not occurring to the probability of occurring, P(Ā):P(A).
The "actual odds in favor" is the ratio of the probability of occurring to the probability of not occurring, P(A):P(Ā).