Dice statistics - preemptive mathhammer

[ildarik]

Hi,

I just did a small analysis of the probability of various outcomes when throwing one or more dice, with a view towards the results that matter to a Mage; I've analyzed up to 6 dice. In each table I'm showing the probability of getting a particular value or better, as well as the probilities for a double-6 and a double-anything.

number of Dice: 1
1+: 100.00 %
2+: 83.33 %
3+: 66.67 %
4+: 50.00 %
5+: 33.33 %
6+: 16.67 %
2x6: 0.00 %
2xX: 0.00 %

number of Dice: 2
2+: 100.00 %
3+: 97.22 %
4+: 91.67 %
5+: 83.33 %
6+: 72.22 %
7+: 58.33 %
8+: 41.67 %
9+: 27.78 %
10+: 16.67 %
11+: 8.33 %
12+: 2.78 %
2x6: 2.78 %
2xX: 16.67 %

number of Dice: 3
3+: 100.00 %
4+: 99.54 %
5+: 98.15 %
6+: 95.37 %
7+: 90.74 %
8+: 83.80 %
9+: 74.07 %
10+: 62.50 %
11+: 50.00 %
12+: 37.50 %
13+: 25.93 %
14+: 16.20 %
15+: 9.26 %
16+: 4.63 %
17+: 1.85 %
18+: 0.46 %
2x6: 7.41 %
2xX: 44.44 %

number of Dice: 4
4+: 100.00 %
5+: 99.92 %
6+: 99.61 %
7+: 98.84 %
8+: 97.30 %
9+: 94.60 %
10+: 90.28 %
11+: 84.10 %
12+: 76.08 %
13+: 66.44 %
14+: 55.63 %
15+: 44.37 %
16+: 33.56 %
17+: 23.92 %
18+: 15.90 %
19+: 9.72 %
20+: 5.40 %
21+: 2.70 %
22+: 1.16 %
23+: 0.39 %
24+: 0.08 %
2x6: 13.19 %
2xX: 72.22 %

number of Dice: 5
5+: 100.00 %
6+: 99.99 %
7+: 99.92 %
8+: 99.73 %
9+: 99.28 %
10+: 98.38 %
11+: 96.76 %
12+: 94.12 %
13+: 90.20 %
14+: 84.80 %
15+: 77.85 %
16+: 69.48 %
17+: 60.03 %
18+: 50.00 %
19+: 39.97 %
20+: 30.52 %
21+: 22.15 %
22+: 15.20 %
23+: 9.80 %
24+: 5.88 %
25+: 3.24 %
26+: 1.62 %
27+: 0.72 %
28+: 0.27 %
29+: 0.08 %
30+: 0.01 %
2x6: 19.62 %
2xX: 90.74 %

number of Dice: 6
6+: 100.00 %
7+: 100.00 % <-- that's a rounding error, of course!
8+: 99.98 %
9+: 99.94 %
10+: 99.82 %
11+: 99.55 %
12+: 99.01 %
13+: 98.03 %
14+: 96.41 %
15+: 93.92 %
16+: 90.35 %
17+: 85.54 %
18+: 79.42 %
19+: 72.06 %
20+: 63.69 %
21+: 54.64 %
22+: 45.36 %
23+: 36.31 %
24+: 27.94 %
25+: 20.58 %
26+: 14.46 %
27+: 9.65 %
28+: 6.08 %
29+: 3.59 %
30+: 1.97 %
31+: 0.99 %
32+: 0.45 %
33+: 0.18 %
34+: 0.06 %
35+: 0.02 %
36+: 0.00 %
2x6: 26.32 %
2xX: 98.46 %

Disclaimer: Don't let knowledge of statistics deter you from praying for the blessings of the fates, using lucky dice, or from crushing misbehaving dice to a fine powder.

[Ruerl Khan]

Disclaimer: Don't let knowledge of statistics deter you from praying for the blessings of the fates, using lucky dice, or from crushing misbehaving dice to a fine powder.

I did that, I crushed a dice that constantly rolled poorly, then I put it down into my box of dice so that the rest of dice could see it as an example of not disappointing me. Naturally, my dice rebelled. -And I learned not to use the d10 dice from white-wolf, the whole angsty thing? Yeah... there was an airbubble just below the "1" making it roll that more often... dice weighted to make you miserable, no wonder their RP system is all about angst.

[Brian Mage]

I'm not sure why more people have not thanked you for this...
Maybe they already have it, i don't, so I'll thank you

Thank You

I'm gonna make this into a .xls pdf... yep i may laminate it too... I love GeekFactor!

[Baerion]

Thanks for this - it's very helpful and I'll keep it near me so that I know how much of a gamble I'm taking every time I roll the dice.

I especially am surprised that a lvl 4 only has a little over 50% chance to get a 25+ for boosted Purple Sun. Perhaps I'll stick with the small one unless I really need the 5" template.

[tiekwando2]

remember that as the number of dice get higher the probability of getting at least 2 six's increases dramatically. So with six dice you can roll a 6,6,1,1,1,1 and still cast the spell. That die role is not included in the 54%. That goes for any roll that has two sixs and totals less than 9 for the other 4 dice. Its not that many, but it does raise the average. If someone has enough time it looks like every permutation of 2 6s, and any combination of 1s and 2s, in any order. Over a total of 6^6 possibilities.

[Baerion]

A good point to keep in mind when using the percentages.

Of course it only applies to situations in which the Dice roll needed is more than 13 with 3 dice, 14 with 4 dice and so on. The rest of the numbers would be accurate if I'm thinking about this correctly as rolling two 6's would mean the roll is successful in the rest of the cases.

Still a very useful tool!

[geoguswrek]

It'd probably be quicker to work out using bayesian statistics.

[ildarik]

remember that as the number of dice get higher the probability of getting at least 2 six's increases dramatically. So with six dice you can roll a 6,6,1,1,1,1 and still cast the spell. That die role is not included in the 54%. That goes for any roll that has two sixs and totals less than 9 for the other 4 dice. Its not that many, but it does raise the average. If someone has enough time it looks like every permutation of 2 6s, and any combination of 1s and 2s, in any order. Over a total of 6^6 possibilities.

Good point. Here's a revised list that shows two percentages for each value. The first is the chance of rolling that much or higher, the second is the chance of rolling that much, or rolling a double-6.

number of Dice: 1
1+: 100.00/100.00 %
2+: 83.33/83.33 %
3+: 66.67/66.67 %
4+: 50.00/50.00 %
5+: 33.33/33.33 %
6+: 16.67/16.67 %
7+: -/0.00 %
2x6: 0.00 %
2xX: 0.00 %

number of Dice: 2
2+: 100.00/100.00 %
3+: 97.22/97.22 %
4+: 91.67/91.67 %
5+: 83.33/83.33 %
6+: 72.22/72.22 %
7+: 58.33/58.33 %
8+: 41.67/41.67 %
9+: 27.78/27.78 %
10+: 16.67/16.67 %
11+: 8.33/8.33 %
12+: 2.78/2.78 %
13+: -/2.78 %
2x6: 2.78 %
2xX: 16.67 %

number of Dice: 3
3+: 100.00/100.00 %
4+: 99.54/99.54 %
5+: 98.15/98.15 %
6+: 95.37/95.37 %
7+: 90.74/90.74 %
8+: 83.80/83.80 %
9+: 74.07/74.07 %
10+: 62.50/62.50 %
11+: 50.00/50.00 %
12+: 37.50/37.50 %
13+: 25.93/25.93 %
14+: 16.20/17.59 %
15+: 9.26/12.04 %
16+: 4.63/8.80 %
17+: 1.85/7.41 %
18+: 0.46/7.41 %
19+: -/7.41 %
2x6: 7.41 %
2xX: 44.44 %

number of Dice: 4
4+: 100.00/100.00 %
5+: 99.92/99.92 %
6+: 99.61/99.61 %
7+: 98.84/98.84 %
8+: 97.30/97.30 %
9+: 94.60/94.60 %
10+: 90.28/90.28 %
11+: 84.10/84.10 %
12+: 76.08/76.08 %
13+: 66.44/66.44 %
14+: 55.63/55.63 %
15+: 44.37/44.83 %
16+: 33.56/34.95 %
17+: 23.92/26.70 %
18+: 15.90/20.52 %
19+: 9.72/16.67 %
20+: 5.40/14.51 %
21+: 2.70/13.50 %
22+: 1.16/13.19 %
23+: 0.39/13.19 %
24+: 0.08/13.19 %
25+: -/13.19 %
2x6: 13.19 %
2xX: 72.22 %

number of Dice: 5
5+: 100.00/100.00 %
6+: 99.99/99.99 %
7+: 99.92/99.92 %
8+: 99.73/99.73 %
9+: 99.28/99.28 %
10+: 98.38/98.38 %
11+: 96.76/96.76 %
12+: 94.12/94.12 %
13+: 90.20/90.20 %
14+: 84.80/84.80 %
15+: 77.85/77.85 %
16+: 69.48/69.61 %
17+: 60.03/60.55 %
18+: 50.00/51.29 %
19+: 39.97/42.54 %
20+: 30.52/35.02 %
21+: 22.15/29.09 %
22+: 15.20/24.85 %
23+: 9.80/22.15 %
24+: 5.88/20.67 %
25+: 3.24/19.96 %
26+: 1.62/19.69 %
27+: 0.72/19.62 %
28+: 0.27/19.62 %
29+: 0.08/19.62 %
30+: 0.01/19.62 %
31+: -/19.62 %
2x6: 19.62 %
2xX: 90.74 %

number of Dice: 6
6+: 100.00/100.00 %
7+: 100.00/100.00 %
8+: 99.98/99.98 %
9+: 99.94/99.94 %
10+: 99.82/99.82 %
11+: 99.55/99.55 %
12+: 99.01/99.01 %
13+: 98.03/98.03 %
14+: 96.41/96.41 %
15+: 93.92/93.92 %
16+: 90.35/90.35 %
17+: 85.54/85.57 %
18+: 79.42/79.58 %
19+: 72.06/72.54 %
20+: 63.69/64.81 %
21+: 54.64/56.89 %
22+: 45.36/49.32 %
23+: 36.31/42.59 %
24+: 27.94/37.05 %
25+: 20.58/32.86 %
26+: 14.46/29.95 %
27+: 9.65/28.12 %
28+: 6.08/27.10 %
29+: 3.59/26.61 %
30+: 1.97/26.40 %
31+: 0.99/26.34 %
32+: 0.45/26.32 %
33+: 0.18/26.32 %
34+: 0.06/26.32 %
35+: 0.02/26.32 %
36+: 0.00/26.32 %
37+: -/26.32 %
2x6: 26.32 %
2xX: 98.46 %

[Thorek the Immortal]

Wow, you've really put some work into this! But why go through all this trouble when someone already has done it

http://www.heresy-online.net/forums/sho ... hp?t=65541

[ildarik]

Wow, you've really put some work into this! But why go through all this trouble when someone already has done it

http://www.heresy-online.net/forums/sho ... hp?t=65541

I don't usually read heresy-online, and I like scripting for fun. It wasn't so bad, about 20 minutes of work. It scales, too, so if someone is interested in the tables for 7 and 8 dice I can do that too

[mishari26]

Actually, I did the same a few weeks ago, and did some graphs with it. I tried to include the probability for ANY successful cast (with double 6's), as well as the probability of a SAFE cast (with no double 6's)

you can see them here:

http://warhammer-empire.com/theforum/in ... ic=34600.0

[Brian Mage]

Oh a revised version - but i already Laminated the first... with smiley faces too

Could you post up the 7 dice one please? useful for the dragon mage...

[ildarik]

Could you post up the 7 dice one please? useful for the dragon mage...

7 and 8 dice, coming up. Took the script 1m13s to crank out

number of Dice: 7
7+: 100.00/100.00 %
8+: 100.00/100.00 %
9+: 100.00/100.00 %
10+: 99.99/99.99 %
11+: 99.96/99.96 %
12+: 99.88/99.88 %
13+: 99.72/99.72 %
14+: 99.39/99.39 %
15+: 98.79/98.79 %
16+: 97.79/97.79 %
17+: 96.21/96.21 %
18+: 93.88/93.89 %
19+: 90.61/90.66 %
20+: 86.28/86.44 %
21+: 80.83/81.25 %
22+: 74.28/75.23 %
23+: 66.78/68.65 %
24+: 58.58/61.88 %
25+: 50.00/55.34 %
26+: 41.42/49.41 %
27+: 33.22/44.38 %
28+: 25.72/40.41 %
29+: 19.17/37.48 %
30+: 13.72/35.51 %
31+: 9.39/34.28 %
32+: 6.12/33.59 %
33+: 3.79/33.24 %
34+: 2.21/33.09 %
35+: 1.21/33.04 %
36+: 0.61/33.02 %
37+: 0.28/33.02 %
38+: 0.12/33.02 %
39+: 0.04/33.02 %
40+: 0.01/33.02 %
41+: 0.00/33.02 %
42+: 0.00/33.02 %
43+: -/33.02 %
2x6: 33.02 %
2xX: 100.00 %

number of Dice: 8
8+: 100.00/100.00 %
9+: 100.00/100.00 %
10+: 100.00/100.00 %
11+: 100.00/100.00 %
12+: 99.99/99.99 %
13+: 99.97/99.97 %
14+: 99.92/99.92 %
15+: 99.82/99.82 %
16+: 99.62/99.62 %
17+: 99.26/99.26 %
18+: 98.63/98.63 %
19+: 97.63/97.63 %
20+: 96.11/96.12 %
21+: 93.93/93.98 %
22+: 90.93/91.07 %
23+: 87.02/87.37 %
24+: 82.11/82.88 %
25+: 76.23/77.72 %
26+: 69.46/72.11 %
27+: 61.98/66.34 %
28+: 54.05/60.71 %
29+: 45.95/55.53 %
30+: 38.02/51.04 %
31+: 30.54/47.38 %
32+: 23.77/44.57 %
33+: 17.89/42.56 %
34+: 12.98/41.22 %
35+: 9.07/40.40 %
36+: 6.07/39.94 %
37+: 3.89/39.70 %
38+: 2.37/39.59 %
39+: 1.37/39.55 %
40+: 0.74/39.54 %
41+: 0.38/39.53 %
42+: 0.18/39.53 %
43+: 0.08/39.53 %
44+: 0.03/39.53 %
45+: 0.01/39.53 %
46+: 0.00/39.53 %
47+: 0.00/39.53 %
48+: 0.00/39.53 %
49+: -/39.53 %
2x6: 39.53 %
2xX: 100.00 %

[tiekwando2]

wow great work!

[Orius]

Are you able to put in the benefits of the + for mage level? how having a level on emage with +1 effects these totals up to a +5 for an archmage with the staff of sorcery?

[tiekwando2]

just subtract from the spell cost. so a boosted purple sun at 25+ to cast. a lvl 4 casts it at 21, lvl 1 casts at 24 etc,

[Orius]

just subtract from the spell cost. so a boosted purple sun at 25+ to cast. a lvl 4 casts it at 21, lvl 1 casts at 24 etc,

good point!

[Jedra]

with a 33% chance of a miscast by your big scary expensive monster mage why would ANYONE roll 7 dice with a dragon mage?

Man, I think I'd crap myself everytime I ever tried something like that...

[Musashi]

When you're stuck in close combat with nothing to lose.

[JudsonCc]

I understand this topic is a year and a half old, but if anyone ever revives it, I'd like to point something out: the "any doubles" odds are, in balance, not very helpful. If we're thinking about it in terms of getting the spell cast with Irrestible Force (using Teclis or BoH), you have to also have gotten the necessary casting value.

As such, for each number of dice, the most helpful information would be three fields. Odds of success / Odds of success OR double 6's / Odds of success that include any doubles.

Best,
Judson