Your example doesn't sound quite right to me. Four-to-one odds means I put up $1,000, the bookmaker or the person I'm betting against puts up $4,000, there is a total pot of $5,000 at stake. The winner gets the whole thing. A total of $5,000, of which part is getting his own money back.
If I win, I get back a total of $5,000--the $1,000 I put up plus the $4,000 the other person put up. It seems to me that I have "won" $4,000 but that I "walk away" with $5,000.
If the horse has a 20% chance of winning, then I have a 20% chance of winning $4,000 and an 80% chance of losing $1,000. So, if the horse has a 20% chance of winning, the bet is fair. In other words, it is possible to convert odds into probabilities.
In the formal mathematical of probability and statistics, I believe the term "odds" is used in Bayesian statistics--it is not just limited to gambling.