The starting point is the following formula, which says that:
The average return on a $ 1 bet (PO) times the probability of winning (PW) minus the probability of losing (PL) is equal to the house edge (E).
(PO x PW) - PL = E
If you know the average return and probability, you can calculate the house edge of the game this way. So if you know the house edge and the probability, you can easily find out the average return of your favourite game.
PO = (E + PL) / PW
What Are The Chances of Winning?
Imagine you are the whale in a casino, and suddenly you feel that the next game will be victorious. The dealer gives himself and you one card face down, from one freshly shuffled deck of cards. The player bets on whether his card will be higher, lower or the same value as the dealer's. The layout of the deck determines the probability of winning, but you must first do a few math operations to get the result.
To find out the probability of this mini-match, suppose the dealer draws any card - say 9. These chances are 4:52.
Only one of the three remaining 9 in the pack will ensure a draw in this game. So our chances are 3:52.
So what is the probability that two 9s hands will be dealt at once? We reach this result by multiplying the two previous values.
(4/52) x (3/52)
This value applies, of course, to cards of all values, from 2 to aces.
So the total probability of two identical unknown cards is 13 x (4/52) x (3/52). This is equal to 1/17 or 5.88 percent.
The opposite value then represents the different cards, i.e. a probability of 16/17 or 94.12 percent.
The chance that a player's card will be higher depends on which card the dealer draws. The chance of any card we have already said is 4:52. Ace is taken as the highest card.
If the dealer draws 2, 12x4 more cards can defeat him. The probabilities of dealer has 2 and that a player will beat him are as follows:
(4/52) x (4 x 12/51) = 7.2389%
If the dealer draws 3, the player's chances of winning are as follows
(4/52) x (4 x 11/51) = 6.6365%
The chances of winning are lower because only 11x4 cards defeat the dealer. Everyone can surely derive further results. And the chance of beating the dealer with his higher cards keeps decreasing, reaching 0 for an ace.
The total probability with two unknown cards is therefore
(4/52) x (48 + 44 + 40 + 36 + 32 + 28 + 24 + 20 + 16 + 12 + 8 + 4 + 0) / 51 = (4/52) x (312/51)
This reduces the probability of winning to 47.06% (8/17). The probability to lose is 52.94% (9/17).
Easy House Edge Calculation
If we know how to calculate the probability, you can calculate the test return for each bet, the value of the house edge, and then decide whether it is reasonable to play such a game. For games like the one we described earlier, it's easier to start with a zero house edge and adjust the payback formula as follows:
PO = PL / PW
For our zero house edge example, we calculate the return as
PO = (16/17) / (1/17) = 16%
However, if we reduce the return to 15%, we get the house edge of the following values:
15 x (1/17) - (16/17) = -1 / 17 or -5.88%
If we reduce the return to 14%, the value of the house edge will rise to
14 x (1/17) - (16/17) = -2/17 or -11.76%
In reality, however, the calculations are far more complicated. In some situations, the return and probability values may be variable. Or the game may have different returns depending on the bets. And games that are easy for players to play even for beginners are not always suitable for this probability analysis. Here you need to use a simulation to determine the chances of different results. But the basic principle is always the same.