To find out the House Edge, we need to compare two things – true odds and payout ratio. While the payout is quite clear, because it is simply quoted by a casino, true odds need to be calculated on the basis of the probability theory.
Despite the gambling industry has its specifics, temptations and dark sides, it is still a regular business as any other. Casinos must make some profit to secure its existence and development. Therefore they (or the casino games) offer lower payouts than those mathematically calculated – so called true or "fair" odds.
How do we stipulate such fair payout? We will take advantage of the knowledge of probability and odds ← here we demonstrate that the odds are just another stipulation or expression of the probability.
Let us demonstrate the calculations on one of the wagers of the old Chinese dice game Sic Bo (Grand Hazard), which is played with three dice. Let us assume that we want to bet on one of the Triples (Raffles), it means that the same number have to occur on all three dice, for example 3 fives.
What is the probability of 3 fives (or any other three same numbers) to come up? Let us count the number of favorable events. It is the only one, since there is the only
1 possibility for 3 fives to be rolled. The number of all possibilities (combinations), which can be created by three dice, is
6 × 6 × 6 = 216. According to the classic definition the probability of rolling 3 fives is
If we stipulate this probability as the Odds, then they are the following: 1:215, i.e. there is only 1 chance to win and 215 chances to lose = 216 chances altogether).
If we mirror this ratio we will get the chance AGAINST us
215:1 and thus the wanted true odds (or fair payout). If the house paid us in this ratio, then it would be achieving zero long-term profit.
However, in terms of the Triple (or Raffle), the house pays
180:1 only. If we compare the true odds and the payout, we will find out that the house pays only
180/215 = 0.837209302 = 83.72%) of the true (fair) odds. The house edge, or we can say the long-term profit or share, results from this difference and thus equals
100% – 83.72% = 16.28%. Perhaps it is easier to arrive at the same conclusion using the expected value.
In this specific case the house edge is extremely big. Still and all other casino games are not bad at all, for instance Roulette gives the house 2.7% edge (or 1.35% in case of even-money bets), Craps 1.4% only. However this example is a great illustration of the importance of the knowledge of the true odds in various casino games and their wagers.