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.
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:
And the probability of "winning" is a / a+b
and the probability of "losing" is b / a+b.
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
Probability to odds:
If the probability of "winning" is n/d,
then the odds in favor of winning are n:d-n
and the odds against winning are d-n:n.
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.
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).