As most CRAPSHOOTERS know, there is a "casino edge" in the game of craps, and it is approximately 1.4% for pass line bets. This means that the casino only has a 1.4% advantage - one of the lowest in the house. But have you ever wondered where this 1.4% figure comes from?
You are probably aware that a pair of dice has 36 combinations. The 2 and 12 can roll one way, the 3 and 11 two ways, the 4 and 10 three ways, the 5 and 9 four ways, the 6 and 8 five ways, and the ubiquitous 7 can roll six ways. On the pass line, you can win with the seven or eleven. The seven can be rolled six ways, and the eleven two ways, so you will win 8 of 36 times on the pass line, or, 2/9 of the time. You will lose on the pass line with a 2, 3 (twice) or 12, so you lose 4 of 36 times, or 1/9 of the time. This means that 1/3 of the time, the bet is decided without a point, as 1/9 (losing) plus 2/9 (winning) equals 3/9 or 1/3. The other 2/3 of the time a point is established.
There are six points - the 4, 5, 6, 8, 9 and 10, with twenty four possible point rolls. If the point is 4 (or 10), as an example, there are 3 ways to roll a 4 (or 10) but 6 ways to roll a 7. Thus, a four (or 10), which rolls three times, compared to the seven, which rolls six times, will roll 3/9 (or 1/3) of the time a seven does. Similarly, with the five and nine, rolling four times, each is 4/10 (or 2/5) as likely to roll as the seven. The six and eight, rolling five times, is 5/11 as likely to roll as the seven. To get the winning percentage then, we simply have to take how often you can possibly win with each point, and multiply it by how often each point occurs.
The casino's edge is the losing percentage minus the winning percentage. After the come-out, there are six point numbers which can be rolled twenty four possible ways (4-3,5-4, 6-5, 8-5, 9-4, 10-3). The four or ten will roll 3/24 (or 1/8) of the time. The five or nine will roll 4/24 (or 1/6) of the time. And the six or eight will roll 5/24 of the time. The chance of winning the four or ten, then, is 1/3 of the time, but only comes up 1/8 of the time. The chance of winning the 5 or 9 is 2/5 of the time, and comes up 1/6 of the time. And, the chance of winning the 6 or 8 is 5/11 of the time, and comes up 5/24 of the time. So, for the 4 and 10, we multiply 1/3 x 1/8 x 2. For the five and nine, we multiply 2/5 x 1/6 x 2. And for the six and eight, we multiply 5/11 x 5/24 x 2. All this adds up to 67/165.
However, we know from before that an actual point is established only 2/3 of the time. 1/3 of the the time you will win or lose without the point being established. And, when no point is established, you'll win 2/3 of the time and lose 1/3 of the time. So, we need to take 2/3 of 67/165 and add it to 1/3 of 2/3, which is 244/495, or the winning percentage, which is 49.3%, or .493. Since the winning percentage is 49.3%, the losing percentage is 50.7% (or .507). We can now finally find the house edge, in percentage, by subtracting the winning percentage (49.3%) from the losing percentage (50.7%). This is 1.4%, the house edge for the pass line with no odds.
It's not important that you know how this 1.4% figure was calculated, but at least now you know! Next time you play, you'll have just a little more knowledge than the next guy!
And, as always, good luck at the tables!