In the case of blackjack the game has a return of anywhere from 98% to 99.9% depending on rule variations, number of decks etc.
Online casinos are fairly cool because their running costs are a fraction of bricks and mortar operations. Fewer staff wages to pay, no utility bills to pay etc etc, plus a busy online casino can accomodate tens of thousands of paying customers at any one time. All this is good news for the cunning punter: the online casino can afford to offer the occasional positive expectation promotion in hope to gain your future loyalty.
For example, Red Lounge casino offered a nice blackjack promotion this weekend:
Dear Red Lounge Casino Member,
This Friday the 13th does not need to be unlucky for you.
No matter what your hand looks like, every time the dealer has a Blackjack, you will get half of your wager back.
Expert players will note that this extremely favourable bonus rule in addition to the existing very favourable blackjack rules gives players a fantastic opportunity to win.
A quick evaluation of this offer goes as follows....
House edge of blackjack at Red Lounge: 0.3%
Frequency of dealer blackjack: 2 x (probability of 10)x(probability of ace).
(There are two ways to order the two cards; either the ace or 10 can come first. The probability that the first card will be a 10 is always 4/13, regardless of the number of decks, because there are 16 out of 52 tens in the deck and 16/52 = 4/13. The probability of an ace, given that a 10 has already been removed from the deck is the number of aces divided by the number of cards left. Let n be the number of decks. There are 4*n aces and 52*n-1 cards left. So for n decks the probability is 2*(4/13)*(4*n/(52*n-1)), which is conveniently about 1 in 21. For example for 6 decks the answer is 2*(4/13)*((4*6)/(52*6-1)) = 192/4043 = 0.047489).
So for every 21 hands we play we get half our stake back. This gives us 0.5/21 back = +2.38%
The game now has a return of 99.7 + 2.38 = 102.8%. It has become a positive expectation game.