The Probabilities of Two Dice Totals
Before you play any dice game it is good to know
the probability of any given total to be thrown. First lets look at
the possibilities of the total of two dice. The table below shows
the six possibilities for die 1 along the left column and the six
possibilities for die 2 along the top column. The body of the table
shows the sum of die 1 and die 2.
| Two dice totals |
| Die 1 |
Die 2 |
| 1 |
2 |
3 |
4 |
5 |
6 |
| 1 |
2 |
3 |
4 |
5 |
6 |
7 |
| 2 |
3 |
4 |
5 |
6 |
7 |
8 |
| 3 |
4 |
5 |
6 |
7 |
8 |
9 |
| 4 |
5 |
6 |
7 |
8 |
9 |
10 |
| 5 |
6 |
7 |
8 |
9 |
10 |
11 |
| 6 |
7 |
8 |
9 |
10 |
11 |
12 |
The colors of the body of the table illustrate the number of ways
to throw each total. The probability of throwing any given total is
the number of ways to throw that total divided by the total number
of combinations (36). In the following table the specific number of
ways to throw each total and the probability of throwing that total
is shown.
| Total |
Number of combinations |
Probability |
| 2 |
1 |
2.78% |
| 3 |
2 |
5.56% |
| 4 |
3 |
8.33% |
| 5 |
4 |
11.11% |
| 6 |
5 |
13.89% |
| 7 |
6 |
16.67% |
| 8 |
5 |
13.89% |
| 9 |
4 |
11.11% |
| 10 |
3 |
8.33% |
| 11 |
2 |
5.56% |
| 12 |
1 |
2.78% |
| Total |
36 |
100% |
The following shows the probability of throwing each total in a
chart format. As the chart shows the closer the total is to 7 the
greater is the probability of it being thrown.
The Field Bet Example
Now that we understand the probability
of throwing each total we can apply this information to the dice
games in the casinos to calculate the house edge. For example
consider the field bet in craps.
This bet pays 1:1 (even money) if the next throw is a 3, 4, 9, 10,
or 11, 2:1 (double the bet) on the 2, and 3:1 (triple the bet) on
the 12. Note that there are 7 totals that win and only 4 that lose
which might cause someone who didn't know better to think it was a
good gamble.
The player's return can be defined as the sum of the products of
the probability of each event and the net return of that event. The
following table shows each possible total, the net return, the
probability of throwing that total, and the average return. The
average return is the product of the net return and the probability.
The player's return is the sum of the average returns.
| Total |
Net return |
Probability |
Average return |
| 2 |
2 |
0.0278 |
0.0556 |
| 3 |
1 |
0.0556 |
0.0556 |
| 4 |
1 |
0.0833 |
0.0833 |
| 5 |
-1 |
0.1111 |
-0.1111 |
| 6 |
-1 |
0.1389 |
-0.1389 |
| 7 |
-1 |
0.1667 |
-0.1667 |
| 8 |
-1 |
0.1389 |
-0.1389 |
| 9 |
1 |
0.1111 |
0.1111 |
| 10 |
1 |
0.0833 |
0.0833 |
| 11 |
1 |
0.0556 |
0.0556 |
| 12 |
3 |
0.0278 |
0.0834 |
| Total |
|
1 |
-0.0278 |
The last row shows the player's return to be -.0278, in other
words for every $1 bet the player can expect to lose 2.78 cents. The
player's loss is the house's gain so the house edge is the product
of -1 and the player's return, in this case 0.0278 or 2.78%.
