Fair value odds are the price an outcome would carry if it were priced at its true probability, with no margin on top. They are the break-even price: at fair value, neither side of the market has an edge.
The term gets used for two related but distinct things, and it is worth keeping them apart.
Fair value from a market. Take a real set of prices, remove the margin, and what remains is the market's own probability estimate expressed as a price. This is what a no-vig calculator produces, and it is the sense meant nine times out of ten. It is "fair" only in the narrow arithmetic sense that the probabilities now sum to 100% — it inherits whatever the market believes, including whatever the market has wrong.
Fair value from a model. Take your own probability estimate and convert it to a price: fair odds = 1 ÷ your probability. This is fair relative to your model, and it says nothing about the market at all. The two numbers are only worth comparing after the first has had its margin removed, which is the entire reason de-vigging exists as a step.
The arithmetic is trivial in both cases. A 50% chance is decimal 2.00, or +100. A 25% chance is 4.00, or +300. What is not trivial is the probability you feed in — and no formula on this page or anywhere else improves the estimate underneath it. A fair value figure is exactly as good as the probability it was derived from, which is why "fair value" is a description of arithmetic, never a claim about accuracy.
One more thing fair value is not: a recommendation. That a market price differs from a fair value figure means the two disagree, and which one is wrong is precisely the open question. We publish a de-vigged CS2 market line as data for you to compare things against, with our own model line still in validation and not yet published.
For the mechanics, see how to remove the vig from odds, or no-vig odds for the closely-related term.