Odds notation and implied probability

Fractional, decimal and moneyline notation are three packagings of one number. Converting them, and finding where the margin in a market sits.

Three notations, one quantity

A price for a bet can be written in at least three ways, and the three are routinely treated as though they carried different information. They do not. Each is a different arithmetic packaging of a single number, the total return per unit staked, and each converts to the others exactly.

• Fractional states profit against stake. 5/1 returns five units of profit for one unit risked, and the stake comes back as well, so the total return is six.
• Decimal states the total return including the stake. The same price is 6.00.
• Moneyline states profit on a hundred units when positive, and the stake required to win a hundred units when negative. The same price is +500.

decimal      =  (fractional numerator / denominator) + 1
implied prob =  1 / decimal
moneyline    =  (decimal - 1) x 100          when decimal >= 2
                -100 / (decimal - 1)         when decimal <  2

Implied probability

The implied probability of a price is the reciprocal of its decimal form. It is the probability at which the bet would break even: at exactly that probability the expected value is zero, above it the bet is positive, below it negative. It is the single most useful conversion in the subject, because it puts prices and probabilities into the same units and makes them comparable.

Worked, for 9/2:

  decimal      = (9 / 2) + 1        =  5.50
  implied prob = 1 / 5.50           =  0.181818  =  18.18 %
  moneyline    = (5.50 - 1) x 100   =  +450

Worked, for 4/5:

  decimal      = (4 / 5) + 1        =  1.80
  implied prob = 1 / 1.80           =  0.555556  =  55.56 %
  moneyline    = -100 / (1.80 - 1)  =  -125

Where the margin lives

Convert every outcome in a market to implied probability and add the results. On a set of prices that carried no margin the sum would be exactly 1. On any market actually offered it is greater than 1, and the excess is the overround. Take a two-outcome market priced at 1.91 on each side.

  1 / 1.91  =  0.523560
  1 / 1.91  =  0.523560
               --------
  sum       =  1.047120       ->  overround = 4.71 %  per market

margin per unit staked, if stakes are placed in
proportion to the implied probabilities:

  0.047120 / 1.047120  =  0.045003  =  4.50 %   per unit staked

Two figures come out of one market, and they are not the same figure. The overround, 4.71 %, is the excess over a whole probability. The margin per unit staked, 4.50 %, is the share of the total amount staked that the prices are expected to retain. Quoting one and labelling it the other is the most common error in this area, which is why every figure on this site is printed with the base it is measured against.

Reading a price as a statement

A price is best read as an assertion about probability plus a margin. A market of prices summing to 1.0471 is asserting a set of probabilities that, once the excess is removed, sum to one; the excess is what the prices retain. This also explains why a price shortening is not evidence about the world: prices move with the amounts staked on each side, because a set of prices that leaves the margin balanced across all outcomes is worth more to the layer than a set that does not.