House edge

The edge is not a fee added to a game and it is not a share taken from winnings. It is the difference between the prize a game pays for a result and the prize that same result would have to pay for the game to break even. Fix the prize below that break-even figure and the edge exists; it is a property of the rules, present before any wheel turns and unaffected by anything that happens afterwards.

The break-even prize

A result that occurs on one trial in n must pay n − 1 units of profit per unit staked for the bet to break even: the single winning trial has to replace the stake lost on each of the other n − 1 trials. On a wheel of thirty-seven pockets a single number occurs once in thirty-seven, so the break-even prize is 36 to 1. The prize actually paid is 35 to 1. The entire edge is that one unit of shortfall, spread across thirty-seven trials.

Single number, 37 pockets, paid 35 to 1

  win   1/37  x  (+35)  =  +0.945946
  lose 36/37  x  ( -1)  =  -0.972973
                          -----------
  expected value        =  -0.027027   per unit staked

  as a fraction: (35 - 36) / 37 = -1/37 = -2.70 %

The same edge on a different bet

An even-money bet on the same wheel covers eighteen of the thirty-seven pockets and pays 1 to 1. The arithmetic differs but the answer does not, because the single pocket lying outside both eighteen-pocket groups is the whole of the shortfall in either case.

Even-money group, 37 pockets, paid 1 to 1

  win  18/37  x  (+1)  =  +0.486486
  lose 19/37  x  (-1)  =  -0.513514
                         -----------
  expected value       =  -0.027027   per unit staked  =  -2.70 %

Adding a second unmatched pocket

A wheel of thirty-eight pockets that still pays 35 to 1 on a single number has two pockets outside the paying structure rather than one, and the edge very nearly doubles.

Single number, 38 pockets, paid 35 to 1

  win   1/38  x  (+35)  =  +0.921053
  lose 37/38  x  ( -1)  =  -0.973684
                          -----------
  expected value        =  -0.052632   per unit staked

  as a fraction: (35 - 37) / 38 = -2/38 = -1/19 = -5.26 %

Two wheels with identical prizes and nearly identical appearance therefore differ by a factor of just under two in what they are expected to retain, and the difference is visible only in the pocket count. This is why an edge cannot be judged from prizes alone: the prize is one half of a comparison, and the count of ways to lose is the other.

What the figure applies to

The edge is stated per unit staked, which means it applies to turnover and not to the amount a player began with. A figure of 2.70 % describes what happens to each unit as it is staked, and a unit that is won back and staked again is counted again. The consequence is arithmetical: expected loss grows in proportion to total amounts staked, without limit, as the count of trials rises.