Relationship between Odds and Probability

It's a simple mathematical relationship, but sometimes I forget.

Odds=p/(1-p)=proportion of success/proportion of failure
p=O/(1+O)=proportion of successes

If there is a probability p of something happening, the the odds (O) is, on average, the number of successes you expect per failure.  High odds correspond to high probabilities and low odds correspond to low probabilities.  As the probability approaches zero, the odds is approximately equal to p.  Probabilities are bound to [0,1], while odds are bound to [0,∞).