Roulette - House edge

The house average or house edge (also called the expected value) is the amount the player loses relative to any bet made, on average. If a player bets on a single number in the American game there is a probability of 1/38 that the player wins 35 times the bet, and a 37/38 chance that the player loses their bet. The expected value is:

-1*37/38 + 35*1/38 = -0.0526 (5.26% house edge)

For European roulette, a single number wins 1/37 and loses 36/37:

-1*36/37 + 35*1/37 = -0.0270 (2.70% house edge)

In roulette the house has the same edge on all other kinds of bets also, because the pay outs are always set as if the zero square(s) did not exist. The only exception are the five numbers bet where the house edge is considerably higher (7.89% on an American wheel) and the 'even money' bets in the some European games, where the house edge is halved because only half the stake is lost when a zero comes up.

The house edge should not be confused with the hold. The hold is the total amount of cash the table changes for chips, minus the chips taken away from the table. It may be significantly more than 5.26%/2.70% of all players money because players are making repeated bets after winning and losing portions of their total money. A player with a certain total amount of money may not win or lose all their money instantly, such that the total of all bets they make will often be greater than the total of the money they actually started with. The house edge applies to each bet made not the total money, which means the player can end up losing significantly more than 5.26% of his starting money.

For example it is likely that a player with $100 making $10 bets on red will be able to bet more than 10 times, because sometimes he wins. He may end up betting a total of 20 times on red. This means the expected value is 20*$10*5.26% = $10.52, over 10% of his money is now in the 'hold' despite the game having a 5.26% house advantage. A player who continually bets until they run out of money will give the house 100% hold.

From Wikipedia's