What is the difference between the margin and the vigorish?

The margin, also known as the overround, is the percentage points over 100% that the total implied probabilities of all possible outcomes add up to. It represents the bookmaker's theoretical advantage if they balance the book perfectly, that is, they have equal amounts bet on all outcomes.

Consider an event with two possible outcomes, x and -x. The odds offered are x: 1.25, -x: 4. These odds imply probabilities of 80% (1 / 1.25) for x and 25% (1 / 4) for -x. The sum of these implied probabilities is 105%, which is 5 percentage points higher than the sum of 100 percentage points that should result if the odds truly were equivalent to the probabilities. This extra 5% is the bookmaker's margin.

The term "vigorish", derived from the Russian word for "winnings", is often used to describe the bookmaker's actual profit on a market. Let's suppose that for the aforementioned odds the total bets collected are: ¤800 for x at 1.25, ¤250 for -x at 4, for a total of ¤1050. If x happens, then the bookmaker pays back ¤1000, that is the ¤800 to the players that gambled for it, plus the prize of ¤200, and keeps ¤50 as profit; if -x happens, the bookmaker pays ¤250 to the players plus ¤750, and keeps also ¤50 as profit.

Although the vigorish is very related to the margin, it is not equivalent to it:

¤50 / ¤1050 ≈ .0476 ≈ 4.76%

The profit of ¤50 is approximately 4.76% from the total collection of ¤1050, which is not the same as the margin of 5 percentage points. However, both can be determined in terms of the other:

vigorish = margin / (1 + margin)
margin = vigorish / (1 - vigorish)

Applied to our example:

vigorish = .05 / (1 + .05) ≈ 4.76%
margin ≈ .0476 / (1 - .0476) ≈ 5%

However, these concepts are often used as if they were interchangeable, and therefore it is important to always take a look at the context in which they are being used, to avoid confusions. Other names also given to these concepts are overround, edge, juice, cut, vig, take, rake... The same precaution must be taken when interpreting these terms in the gambling literature, which tends to be very informal and imprecise.