Game Probabilitiestutorial Release 1
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Layman's Tutorial on Dice probabilities Copyright J.D. Neal 2008-2009 All Rights Reserved The layman's simple way of calculating the probabilities of dice rolls is to use a piece of graph paper or spreadsheet to count out the combinations, then convert them to a percentage. For example, below is 1d6 + 1d8 to generate numbers from 2 to 14:
1d8
1d6 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
7
8
9
10
11
12
13
8
9
10
11
12
13
14
The way the grid works: You write out the possible numbers on 1d6 up top, and on 1d8 to the left, then go through and add them up, filling in each square. When rolling 1d6 and adding it to 1d8, when you roll a 1 on both numbers, the total is 2. When you roll a 1 on 1d6 and 2 on 1d8, the total is 3. When you roll a 4 on 1d6 and a 5 on 1d8, the total is 9, and so on. Note that there are 48 combinations (which also happens to be 6 x 8). Go through and count them out: 2 occurs once, 3 occurs twice, all the way up to 10 which occurs 5 times, then 11 which occurs 4 times, 12 occurs 3 times, 13 occurs twice, and 14 once.
Number 2 3 4 5 6 7 8 9 10 11 12 13
Combinations 1 2 3 4 5 6 6 6 5 4 3 2
Fraction 1/48 2/48 3/48 4/48 5/48 6/48 6/48 6/48 5/48 4/48 3/48 2/48
Percent 2.08% 4.17% 6.25% 8.33% 10.42% 12.50% 12.50% 12.50% 10.42% 8.33% 6.25% 4.17%
14
1
1/48
2.08%
Totals:
48
48/48
100.00%
Compare that to a roll of 1d4 + 1d10 to create numbers from 2 to 14. There are 40 combinations (which is also 4 x 10).
1d4 1d10
1
2
3
4
1
2
3
4
5
2
3
4
5
6
3
4
5
6
7
4
5
6
7
8
5
6
7
8
9
6
7
8
9
10
7
8
9
10
11
8
9
10
11
12
9
10
11
12
13
10
11
12
13
14
Number
Combinations
Fraction
Percent
2
1
1/40
2.50%
3
2
2/40
5.00%
4
3
3/40
7.50%
5
4
4/40
10.00%
6
4
4/40
10.00%
7
4
4/40
10.00%
8
4
4/40
10.00%
9
4
4/40
10.00%
10
4
4/40
10.00%
11
4
4/40
10.00%
12
3
3/40
7.50%
13
2
2/40
5.00%
14
1
1/40
2.50%
Totals:
40
40/40
100.00%
This is an analysis of 2d6 (1d6+1d6).
1d6
1
2
1d6 3
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
4
5
6
2d6 # 2
Occurences 1
Fraction 1/36
Percentage 2.78%
Cumulative % 2.78%
3
2
2/36
5.56%
8.33%
4
3
3/36
8.33%
16.67%
5
4
4/36
11.11%
27.78%
6
5
5/36
13.89%
41.67%
7
6
6/36
16.67%
58.33%
8
5
5/36
13.89%
72.22%
9
4
4/36
11.11%
83.33%
10
3
3/36
8.33%
91.67%
11
2
2/36
5.56%
97.22%
12
1
1/36
2.78%
100.00%
36
Here's an analysis of 2d10 (1d10+1d10).
100.00%
1d10
1
1 2
2 3
3 4
1d10 4 5
2
3
4
5
6
7
8
9
10
11
12
3
4
5
6
7
8
9
10
11
12
13
4
5
6
7
8
9
10
11
12
13
14
5
6
7
8
9
10
11
12
13
14
15
6
7
8
9
10
11
12
13
14
15
16
7
8
9
10
11
12
13
14
15
16
17
8
9
10
11
12
13
14
15
16
17
18
9
10
11
12
13
14
15
16
17
18
19
10
11
12
13
14
15
16
17
18
19
20
5 6
6 7
7 8
8 9
9 10
10 11
2d10 # 2
Occurrences 1
Fraction 1/100
Percentage 1.00%
Cumulative % 1.00%
3
2
2/100
2.00%
3.00%
4
3
3/100
3.00%
6.00%
5
4
4/100
4.00%
10.00%
6
5
5/100
5.00%
15.00%
7
6
6/100
6.00%
21.00%
8
7
7/100
7.00%
28.00%
9
8
8/100
8.00%
36.00%
10
9
9/100
9.00%
45.00%
11
10
10/100
10.00%
55.00%
12
9
9/100
9.00%
64.00%
13
8
8/100
8.00%
72.00%
14
7
7/100
7.00%
79.00%
15
6
6/100
6.00%
85.00%
16
5
5/100
5.00%
90.00%
17
4
4/100
4.00%
94.00%
18
3
3/100
3.00%
97.00%
19
2
2/100
2.00%
99.00%
20
1
1/100
1.00%
100.00%
100
100.00%
When using three dice, add two up to make one grid, then add the number of the other dice to the grids. The following is an analysis of d4+d6+d8 to make numbers 3 - 18. Save the smallest dice (if there is one) for the final step, so first add d6+d8. 1d6 1d8
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
7
8
9
10
11
12
13
8
9
10
11
12
13
14
A d4 creates the numbers 1 to 4, so make 4 more grids marked 1 to 4 and add the numbers to the main grid. 1 on 1d4
2 on 1d4
3
4
5
6
7
8
4
5
6
7
8
9
4
5
6
7
8
9
5
6
7
8
9
10
5
6
7
8
9
10
6
7
8
9
10
11
6
7
8
9
10
11
7
8
9
10
11
12
7
8
9
10
11
12
8
9
10
11
12
13
8
9
10
11
12
13
9
10
11
12
13
14
9
10
11
12
13
14
10
11
12
13
14
15
10
11
12
13
14
15
11
12
13
14
15
16
3 on 1d4
4 on 1d4
5
6
7
8
9
10
6
7
8
9
10
11
6
7
8
9
10
11
7
8
9
10
11
12
7
8
9
10
11
12
8
9
10
11
12
13
8
9
10
11
12
13
9
10
11
12
13
14
9
10
11
12
13
14
10
11
12
13
14
15
10
11
12
13
14
15
11
12
13
14
15
16
11
12
13
14
15
16
12
13
14
15
16
17
12
13
14
15
16
17
13
14
15
16
17
18
The following is an example of analyzing 3d6. The dice are all the same size, so make one main grid of d6 + d6, then six other grids where the numbers 1 to 6 are added to the main grid. 1d6
1d6
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
1 on 1d6
2 on 1d6
3
4
5
6
7
8
4
5
6
7
8
9
4
5
6
7
8
9
5
6
7
8
9
10
5
6
7
8
9
10
6
7
8
9
10
11
6
7
8
9
10
11
7
8
9
10
11
12
7
8
9
10
11
12
8
9
10
11
12
13
8
9
10
11
12
13
9
10
11
12
13
14
3 on 1d6
4 on 1d6
5
6
7
8
9
10
6
7
8
9
10
11
6
7
8
9
10
11
7
8
9
10
11
12
7
8
9
10
11
12
8
9
10
11
12
13
8
9
10
11
12
13
9
10
11
12
13
14
9
10
11
12
13
14
10
11
12
13
14
15
10
11
12
13
14
15
11
12
13
14
15
16
5 on 1d6
6 on 1d6
7
8
9
10
11
12
8
9
10
11
12
13
8
9
10
11
12
13
9
10
11
12
13
14
9
10
11
12
13
14
10
11
12
13
14
15
10
11
12
13
14
15
11
12
13
14
15
16
11
12
13
14
15
16
12
13
14
15
16
17
12
13
14
15
16
17
13
14
15
16
17
18
Counting the combinations of d4+d6+d8 and 3d6 (d6+d6+d6) you get the following. Note that there are 4 x 6 x 8 = 192 combinations for d4+d6+d8 and 6 x 6 x 6 = 216 for 3d6. d4+d6+d8
3d6
#
Combinations
Fraction
Percent
Combinations
Fraction
Percent
3
1
1/192
0.52%
1
1/216
0.46%
4
3
3/192
1.56%
3
3/216
1.39%
5
6
6/192
3.13%
6
6/216
2.78%
6
10
10/192
5.21%
10
10/216
4.63%
7
14
14/192
7.29%
15
15/216
6.94%
8
18
18/192
9.38%
21
21/216
9.72%
9
21
21/192
10.94%
25
25/216
11.57%
10
23
23/192
11.98%
27
27/216
12.50%
11
23
23/192
11.98%
27
27/216
12.50%
12
21
21/192
10.94%
25
25/216
11.57%
13
18
18/192
9.38%
21
21/216
9.72%
14
14
14/192
7.29%
15
15/216
6.94%
15
10
10/192
5.21%
10
10/216
4.63%
16
6
6/192
3.13%
6
6/216
2.78%
17
3
3/192
1.56%
3
3/216
1.39%
18
1
1/192
0.52%
1
1/216
0.46%
1
216
192
Odds I grew up confusing the term probabilities with the term odds. Odds is the chance of something happening compared to it not happening. If you roll 1d6 and are trying to get a 5 or 6, the odds are 4 to 2 (2 to 1) against it. If you are trying to get 4, 5, or 6, the odds are even (3 to 3, or 1 to 1, or 50-50). If you are trying to get 2 to 6, the odds are 5 to 1 in your favor.
1d8
1d6 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
7
8
9
10
11
12
13
8
9
10
11
12
13
14
The way the grid works: You write out the possible numbers on 1d6 up top, and on 1d8 to the left, then go through and add them up, filling in each square. When rolling 1d6 and adding it to 1d8, when you roll a 1 on both numbers, the total is 2. When you roll a 1 on 1d6 and 2 on 1d8, the total is 3. When you roll a 4 on 1d6 and a 5 on 1d8, the total is 9, and so on. Note that there are 48 combinations (which also happens to be 6 x 8). Go through and count them out: 2 occurs once, 3 occurs twice, all the way up to 10 which occurs 5 times, then 11 which occurs 4 times, 12 occurs 3 times, 13 occurs twice, and 14 once.
Number 2 3 4 5 6 7 8 9 10 11 12 13
Combinations 1 2 3 4 5 6 6 6 5 4 3 2
Fraction 1/48 2/48 3/48 4/48 5/48 6/48 6/48 6/48 5/48 4/48 3/48 2/48
Percent 2.08% 4.17% 6.25% 8.33% 10.42% 12.50% 12.50% 12.50% 10.42% 8.33% 6.25% 4.17%
14
1
1/48
2.08%
Totals:
48
48/48
100.00%
Compare that to a roll of 1d4 + 1d10 to create numbers from 2 to 14. There are 40 combinations (which is also 4 x 10).
1d4 1d10
1
2
3
4
1
2
3
4
5
2
3
4
5
6
3
4
5
6
7
4
5
6
7
8
5
6
7
8
9
6
7
8
9
10
7
8
9
10
11
8
9
10
11
12
9
10
11
12
13
10
11
12
13
14
Number
Combinations
Fraction
Percent
2
1
1/40
2.50%
3
2
2/40
5.00%
4
3
3/40
7.50%
5
4
4/40
10.00%
6
4
4/40
10.00%
7
4
4/40
10.00%
8
4
4/40
10.00%
9
4
4/40
10.00%
10
4
4/40
10.00%
11
4
4/40
10.00%
12
3
3/40
7.50%
13
2
2/40
5.00%
14
1
1/40
2.50%
Totals:
40
40/40
100.00%
This is an analysis of 2d6 (1d6+1d6).
1d6
1
2
1d6 3
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
4
5
6
2d6 # 2
Occurences 1
Fraction 1/36
Percentage 2.78%
Cumulative % 2.78%
3
2
2/36
5.56%
8.33%
4
3
3/36
8.33%
16.67%
5
4
4/36
11.11%
27.78%
6
5
5/36
13.89%
41.67%
7
6
6/36
16.67%
58.33%
8
5
5/36
13.89%
72.22%
9
4
4/36
11.11%
83.33%
10
3
3/36
8.33%
91.67%
11
2
2/36
5.56%
97.22%
12
1
1/36
2.78%
100.00%
36
Here's an analysis of 2d10 (1d10+1d10).
100.00%
1d10
1
1 2
2 3
3 4
1d10 4 5
2
3
4
5
6
7
8
9
10
11
12
3
4
5
6
7
8
9
10
11
12
13
4
5
6
7
8
9
10
11
12
13
14
5
6
7
8
9
10
11
12
13
14
15
6
7
8
9
10
11
12
13
14
15
16
7
8
9
10
11
12
13
14
15
16
17
8
9
10
11
12
13
14
15
16
17
18
9
10
11
12
13
14
15
16
17
18
19
10
11
12
13
14
15
16
17
18
19
20
5 6
6 7
7 8
8 9
9 10
10 11
2d10 # 2
Occurrences 1
Fraction 1/100
Percentage 1.00%
Cumulative % 1.00%
3
2
2/100
2.00%
3.00%
4
3
3/100
3.00%
6.00%
5
4
4/100
4.00%
10.00%
6
5
5/100
5.00%
15.00%
7
6
6/100
6.00%
21.00%
8
7
7/100
7.00%
28.00%
9
8
8/100
8.00%
36.00%
10
9
9/100
9.00%
45.00%
11
10
10/100
10.00%
55.00%
12
9
9/100
9.00%
64.00%
13
8
8/100
8.00%
72.00%
14
7
7/100
7.00%
79.00%
15
6
6/100
6.00%
85.00%
16
5
5/100
5.00%
90.00%
17
4
4/100
4.00%
94.00%
18
3
3/100
3.00%
97.00%
19
2
2/100
2.00%
99.00%
20
1
1/100
1.00%
100.00%
100
100.00%
When using three dice, add two up to make one grid, then add the number of the other dice to the grids. The following is an analysis of d4+d6+d8 to make numbers 3 - 18. Save the smallest dice (if there is one) for the final step, so first add d6+d8. 1d6 1d8
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
7
8
9
10
11
12
13
8
9
10
11
12
13
14
A d4 creates the numbers 1 to 4, so make 4 more grids marked 1 to 4 and add the numbers to the main grid. 1 on 1d4
2 on 1d4
3
4
5
6
7
8
4
5
6
7
8
9
4
5
6
7
8
9
5
6
7
8
9
10
5
6
7
8
9
10
6
7
8
9
10
11
6
7
8
9
10
11
7
8
9
10
11
12
7
8
9
10
11
12
8
9
10
11
12
13
8
9
10
11
12
13
9
10
11
12
13
14
9
10
11
12
13
14
10
11
12
13
14
15
10
11
12
13
14
15
11
12
13
14
15
16
3 on 1d4
4 on 1d4
5
6
7
8
9
10
6
7
8
9
10
11
6
7
8
9
10
11
7
8
9
10
11
12
7
8
9
10
11
12
8
9
10
11
12
13
8
9
10
11
12
13
9
10
11
12
13
14
9
10
11
12
13
14
10
11
12
13
14
15
10
11
12
13
14
15
11
12
13
14
15
16
11
12
13
14
15
16
12
13
14
15
16
17
12
13
14
15
16
17
13
14
15
16
17
18
The following is an example of analyzing 3d6. The dice are all the same size, so make one main grid of d6 + d6, then six other grids where the numbers 1 to 6 are added to the main grid. 1d6
1d6
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
1 on 1d6
2 on 1d6
3
4
5
6
7
8
4
5
6
7
8
9
4
5
6
7
8
9
5
6
7
8
9
10
5
6
7
8
9
10
6
7
8
9
10
11
6
7
8
9
10
11
7
8
9
10
11
12
7
8
9
10
11
12
8
9
10
11
12
13
8
9
10
11
12
13
9
10
11
12
13
14
3 on 1d6
4 on 1d6
5
6
7
8
9
10
6
7
8
9
10
11
6
7
8
9
10
11
7
8
9
10
11
12
7
8
9
10
11
12
8
9
10
11
12
13
8
9
10
11
12
13
9
10
11
12
13
14
9
10
11
12
13
14
10
11
12
13
14
15
10
11
12
13
14
15
11
12
13
14
15
16
5 on 1d6
6 on 1d6
7
8
9
10
11
12
8
9
10
11
12
13
8
9
10
11
12
13
9
10
11
12
13
14
9
10
11
12
13
14
10
11
12
13
14
15
10
11
12
13
14
15
11
12
13
14
15
16
11
12
13
14
15
16
12
13
14
15
16
17
12
13
14
15
16
17
13
14
15
16
17
18
Counting the combinations of d4+d6+d8 and 3d6 (d6+d6+d6) you get the following. Note that there are 4 x 6 x 8 = 192 combinations for d4+d6+d8 and 6 x 6 x 6 = 216 for 3d6. d4+d6+d8
3d6
#
Combinations
Fraction
Percent
Combinations
Fraction
Percent
3
1
1/192
0.52%
1
1/216
0.46%
4
3
3/192
1.56%
3
3/216
1.39%
5
6
6/192
3.13%
6
6/216
2.78%
6
10
10/192
5.21%
10
10/216
4.63%
7
14
14/192
7.29%
15
15/216
6.94%
8
18
18/192
9.38%
21
21/216
9.72%
9
21
21/192
10.94%
25
25/216
11.57%
10
23
23/192
11.98%
27
27/216
12.50%
11
23
23/192
11.98%
27
27/216
12.50%
12
21
21/192
10.94%
25
25/216
11.57%
13
18
18/192
9.38%
21
21/216
9.72%
14
14
14/192
7.29%
15
15/216
6.94%
15
10
10/192
5.21%
10
10/216
4.63%
16
6
6/192
3.13%
6
6/216
2.78%
17
3
3/192
1.56%
3
3/216
1.39%
18
1
1/192
0.52%
1
1/216
0.46%
1
216
192
Odds I grew up confusing the term probabilities with the term odds. Odds is the chance of something happening compared to it not happening. If you roll 1d6 and are trying to get a 5 or 6, the odds are 4 to 2 (2 to 1) against it. If you are trying to get 4, 5, or 6, the odds are even (3 to 3, or 1 to 1, or 50-50). If you are trying to get 2 to 6, the odds are 5 to 1 in your favor.
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