# 2014-06-24 Emacs and Dice

If I roll one of the five standard gaming dice (d4, d6, d8, d10, d12) and you also roll one of those, what are my chances of rolling higher, equal or lower than you, for each possible combination of dice.﻿

I decided to write a little program to simply count it since I was unable to figure it out by thinking it through.

Update: Now that I’ve looked at the data, the formula is obvious. Assume x is the lower die:

x winning `(x(x-1))/2/(xy) = (x-1)/(2y)`
tie `x/(xy) = 1/y`
y winning `((x(x-1))/2 + x(y-x))/(xy) = 1 - ((x+1)/(2y))`
```(dolist (die-1 '(4 6 8 10 12))
(dolist (die-2 '(4 6 8 10 12))
(unless (< die-2 die-1)
(let ((a 0) (b 0) (n 0))
(insert (format "d%d vs. d%d" die-1 die-2))
(newline)
(newline)
(insert "     ")
(dotimes (roll-1 die-1)
(insert (format " %4d" (1+ roll-1))))
(newline)
(dotimes (roll-2 die-2)
(insert (format " %4d" (1+ roll-2)))
(dotimes (roll-1 die-1)
(insert "    "
(cond ((> roll-1 roll-2)
(incf a)
"A")
((= roll-1 roll-2)
(incf n)
"-")
(t
(incf b)
"B"))))
(newline))
(newline)
(let ((total (+ a b n)))
(insert (format "A: %d / %d = %4.3f\n" a total (/ a 1.0 total)))
(insert (format "B: %d / %d = %4.3f\n" b total (/ b 1.0 total)))
(insert (format "-: %d / %d = %4.3f\n" n total (/ n 1.0 total))))
(newline)
(newline)))))

d4 vs. d4

1    2    3    4
1    -    A    A    A
2    B    -    A    A
3    B    B    -    A
4    B    B    B    -

A: 6 / 16 = 0.375
B: 6 / 16 = 0.375
-: 4 / 16 = 0.250

d4 vs. d6

1    2    3    4
1    -    A    A    A
2    B    -    A    A
3    B    B    -    A
4    B    B    B    -
5    B    B    B    B
6    B    B    B    B

A: 6 / 24 = 0.250
B: 14 / 24 = 0.583
-: 4 / 24 = 0.167

d4 vs. d8

1    2    3    4
1    -    A    A    A
2    B    -    A    A
3    B    B    -    A
4    B    B    B    -
5    B    B    B    B
6    B    B    B    B
7    B    B    B    B
8    B    B    B    B

A: 6 / 32 = 0.188
B: 22 / 32 = 0.688
-: 4 / 32 = 0.125

d4 vs. d10

1    2    3    4
1    -    A    A    A
2    B    -    A    A
3    B    B    -    A
4    B    B    B    -
5    B    B    B    B
6    B    B    B    B
7    B    B    B    B
8    B    B    B    B
9    B    B    B    B
10    B    B    B    B

A: 6 / 40 = 0.150
B: 30 / 40 = 0.750
-: 4 / 40 = 0.100

d4 vs. d12

1    2    3    4
1    -    A    A    A
2    B    -    A    A
3    B    B    -    A
4    B    B    B    -
5    B    B    B    B
6    B    B    B    B
7    B    B    B    B
8    B    B    B    B
9    B    B    B    B
10    B    B    B    B
11    B    B    B    B
12    B    B    B    B

A: 6 / 48 = 0.125
B: 38 / 48 = 0.792
-: 4 / 48 = 0.083

d6 vs. d6

1    2    3    4    5    6
1    -    A    A    A    A    A
2    B    -    A    A    A    A
3    B    B    -    A    A    A
4    B    B    B    -    A    A
5    B    B    B    B    -    A
6    B    B    B    B    B    -

A: 15 / 36 = 0.417
B: 15 / 36 = 0.417
-: 6 / 36 = 0.167

d6 vs. d8

1    2    3    4    5    6
1    -    A    A    A    A    A
2    B    -    A    A    A    A
3    B    B    -    A    A    A
4    B    B    B    -    A    A
5    B    B    B    B    -    A
6    B    B    B    B    B    -
7    B    B    B    B    B    B
8    B    B    B    B    B    B

A: 15 / 48 = 0.312
B: 27 / 48 = 0.562
-: 6 / 48 = 0.125

d6 vs. d10

1    2    3    4    5    6
1    -    A    A    A    A    A
2    B    -    A    A    A    A
3    B    B    -    A    A    A
4    B    B    B    -    A    A
5    B    B    B    B    -    A
6    B    B    B    B    B    -
7    B    B    B    B    B    B
8    B    B    B    B    B    B
9    B    B    B    B    B    B
10    B    B    B    B    B    B

A: 15 / 60 = 0.250
B: 39 / 60 = 0.650
-: 6 / 60 = 0.100

d6 vs. d12

1    2    3    4    5    6
1    -    A    A    A    A    A
2    B    -    A    A    A    A
3    B    B    -    A    A    A
4    B    B    B    -    A    A
5    B    B    B    B    -    A
6    B    B    B    B    B    -
7    B    B    B    B    B    B
8    B    B    B    B    B    B
9    B    B    B    B    B    B
10    B    B    B    B    B    B
11    B    B    B    B    B    B
12    B    B    B    B    B    B

A: 15 / 72 = 0.208
B: 51 / 72 = 0.708
-: 6 / 72 = 0.083

d8 vs. d8

1    2    3    4    5    6    7    8
1    -    A    A    A    A    A    A    A
2    B    -    A    A    A    A    A    A
3    B    B    -    A    A    A    A    A
4    B    B    B    -    A    A    A    A
5    B    B    B    B    -    A    A    A
6    B    B    B    B    B    -    A    A
7    B    B    B    B    B    B    -    A
8    B    B    B    B    B    B    B    -

A: 28 / 64 = 0.438
B: 28 / 64 = 0.438
-: 8 / 64 = 0.125

d8 vs. d10

1    2    3    4    5    6    7    8
1    -    A    A    A    A    A    A    A
2    B    -    A    A    A    A    A    A
3    B    B    -    A    A    A    A    A
4    B    B    B    -    A    A    A    A
5    B    B    B    B    -    A    A    A
6    B    B    B    B    B    -    A    A
7    B    B    B    B    B    B    -    A
8    B    B    B    B    B    B    B    -
9    B    B    B    B    B    B    B    B
10    B    B    B    B    B    B    B    B

A: 28 / 80 = 0.350
B: 44 / 80 = 0.550
-: 8 / 80 = 0.100

d8 vs. d12

1    2    3    4    5    6    7    8
1    -    A    A    A    A    A    A    A
2    B    -    A    A    A    A    A    A
3    B    B    -    A    A    A    A    A
4    B    B    B    -    A    A    A    A
5    B    B    B    B    -    A    A    A
6    B    B    B    B    B    -    A    A
7    B    B    B    B    B    B    -    A
8    B    B    B    B    B    B    B    -
9    B    B    B    B    B    B    B    B
10    B    B    B    B    B    B    B    B
11    B    B    B    B    B    B    B    B
12    B    B    B    B    B    B    B    B

A: 28 / 96 = 0.292
B: 60 / 96 = 0.625
-: 8 / 96 = 0.083

d10 vs. d10

1    2    3    4    5    6    7    8    9   10
1    -    A    A    A    A    A    A    A    A    A
2    B    -    A    A    A    A    A    A    A    A
3    B    B    -    A    A    A    A    A    A    A
4    B    B    B    -    A    A    A    A    A    A
5    B    B    B    B    -    A    A    A    A    A
6    B    B    B    B    B    -    A    A    A    A
7    B    B    B    B    B    B    -    A    A    A
8    B    B    B    B    B    B    B    -    A    A
9    B    B    B    B    B    B    B    B    -    A
10    B    B    B    B    B    B    B    B    B    -

A: 45 / 100 = 0.450
B: 45 / 100 = 0.450
-: 10 / 100 = 0.100

d10 vs. d12

1    2    3    4    5    6    7    8    9   10
1    -    A    A    A    A    A    A    A    A    A
2    B    -    A    A    A    A    A    A    A    A
3    B    B    -    A    A    A    A    A    A    A
4    B    B    B    -    A    A    A    A    A    A
5    B    B    B    B    -    A    A    A    A    A
6    B    B    B    B    B    -    A    A    A    A
7    B    B    B    B    B    B    -    A    A    A
8    B    B    B    B    B    B    B    -    A    A
9    B    B    B    B    B    B    B    B    -    A
10    B    B    B    B    B    B    B    B    B    -
11    B    B    B    B    B    B    B    B    B    B
12    B    B    B    B    B    B    B    B    B    B

A: 45 / 120 = 0.375
B: 65 / 120 = 0.542
-: 10 / 120 = 0.083

d12 vs. d12

1    2    3    4    5    6    7    8    9   10   11   12
1    -    A    A    A    A    A    A    A    A    A    A    A
2    B    -    A    A    A    A    A    A    A    A    A    A
3    B    B    -    A    A    A    A    A    A    A    A    A
4    B    B    B    -    A    A    A    A    A    A    A    A
5    B    B    B    B    -    A    A    A    A    A    A    A
6    B    B    B    B    B    -    A    A    A    A    A    A
7    B    B    B    B    B    B    -    A    A    A    A    A
8    B    B    B    B    B    B    B    -    A    A    A    A
9    B    B    B    B    B    B    B    B    -    A    A    A
10    B    B    B    B    B    B    B    B    B    -    A    A
11    B    B    B    B    B    B    B    B    B    B    -    A
12    B    B    B    B    B    B    B    B    B    B    B    -

A: 66 / 144 = 0.458
B: 66 / 144 = 0.458
-: 12 / 144 = 0.083```

Tags:

I thought you drew the visualizations by hand. I didn’t think your little piece of Lisp code drew them. Three cheers for simulations with Lisp!

Aaron S. Hawley 2014-06-24 23:05 UTC

Thanks.

AlexSchroeder 2014-06-25 10:05 UTC

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