Game ProbabilitiesTutorial Release 1

8/14/2019 Game ProbabilitiesTutorial Release 1

http://slidepdf.com/reader/full/game-probabilitiestutorial-release-1 1/9

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 is1d6 + 1d8 to generate numbers from 2 to 14:

1d6

1d8 1 2 3 4 5 6

1 2 3 4 5 6 7

2 3 4 5 6 7 83 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 youroll 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 Combinations Fraction Percent

2 1 1/48 2.08%

3 2 2/48 4.17%

4 3 3/48 6.25%5 4 4/48 8.33%

6 5 5/48 10.42%7 6 6/48 12.50%8 6 6/48 12.50%

9 6 6/48 12.50%

10 5 5/48 10.42%11 4 4/48 8.33%

12 3 3/48 6.25%

13 2 2/48 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 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

1d6 4 5 6 7 8 9 10

5 6 7 8 9 10 11

6 7 8 9 10 11 12

2d6

# Occurences Fraction Percentage Cumulative %

2 1 1/36 2.78% 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%

36 100.00%

Here's an analysis of 2d10 (1d10+1d10).



1d10

1 2 3 4 5 6 7 8 9 10

1 2 3 4 5 6 7 8 9 10 11

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 141d10 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



2d10

# Occurrences Fraction Percentage Cumulative %

2 1 1/100 1.00% 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

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

1d6 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%

192 1 216

Odds

I grew up confusing the term probabilities with the term odds. Odds is the chance of somethinghappening 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.

Documents

Game ProbabilitiesTutorial Release 1