Dice probability help (again!)

[Morrus]

I still struggle with grasping dice probabilities. I'm hoping somebody who finds it easier can help out.

Given pools of 5d6, 6d6, 7d6....15d6 what is the probability of rolling (a) two or more sixes and (b) three or more sixes?

I tried modelling it in a spreadsheet, but got confused!

 

[Reinhart]

I'm a statistician so I could just tell you the answer? But I think showing you how to get it might serve you better as a game designer. What you are attempting to do is just a series of binomials. Excel can actually do this for you quite easily:

http://www.ehow.com/how_8231796_use-excel-binomial-probability.html

The basic argument is = 1-BINOMDIST(1,X,0.166666667,TRUE)

This gives you two or more successes. Replace X with your dice pool of 5, 6, etc. If you want three or more successes then change the 1 before the X to 2.

If you don't have excel, here's an online binomial calculator I can walk you through.

http://stattrek.com/online-calculator/binomial.aspx

The probability on a single success is 1/6 or roughly .1666667.
The number of trials is your dice pool size.
The number of successes in your given example is 2.
The probability that you're looking for is "Cumulative Probability: P(X > 2)"

To find the probability for 3 or more successes, just change the number of successes to 3.


Edit: Added more information and clarified the Excel syntax.

 

[pemerton]

Given pools of 5d6, 6d6, 7d6....15d6 what is the probability of rolling (a) two or more sixes and (b) three or more sixes?

I tried modelling it in a spreadsheet, but got confused!

This is fairly simple to do even with pen-and-paper and high school combinatorics.

With 5d6, there are 6^5 (=216*36) outcomes. That's your denominator.

You're interested in all the outcomes that don't have either (1) no sixes or (2) one six.

(1) The options with no sixes are when a 5 in 6 chance comes up on all 5 dice, which is 5^5 = (125*25).

(2) Each option with one six is 5^4 * 1. But with 5 dice, there are 5C1 such options (I'm not quite sure how I'm meant to do combinatoric notation in this medium) = 5!/(4!*1!) = 5. So overall, the number of such options is 5*5.

Add (1) and (2) to get your numerator:

5^5+5^5 = 6250

So the probability of not getting either 0 sixes or 1 six =

1 - (6250/216*36)
= (7776-6250)/7776
= 1526/7776

which is approximately one-fifth (at this point a calculator can help!).

Similar maths will work out the same thing for bigger dice pools, and for more than two requires sixes.

 

[Reinhart]

[MENTION=42582]pemerton[/MENTION]'s walk-through of the binomial model is perfect. I just figured if you're not math inclined then a couple of simple tools would suffice. I've checked on Google spreadsheets, and they have the exact same syntax as Excel when it comes to binomial distributions. https://support.google.com/docs/answer/3093987?hl=en

So you don't even need Microsoft Office to quickly calculate these probabilities. Just a modern web browser.

 

[billd91]

http://stattrek.com/online-calculator/binomial.aspx

The probability on a single success is 1/6 or roughly .1666667.
The number of trials is your dice pool size.
The number of successes in your given example is 2.
The probability that you're looking for is "Cumulative Probability: P(X > 2)"

To find the probability for 3 or more successes, just change the number of successes to 3.

I'm definitely bookmarking that site!

 

[Morrus]

I'm a statistician so I could just tell you the answer? But I think showing you how to get it might serve you better as a game designer. What you are attempting to do is just a series of binomials. Excel can actually do this for you quite easily:

http://www.ehow.com/how_8231796_use-excel-binomial-probability.html

The basic argument is = 1-BINOMDIST(1,X,0.166666667,TRUE)

This gives you two or more successes. Replace X with your dice pool of 5, 6, etc. If you want three or more successes then change the 1 before the X to 2.

If you don't have excel, here's an online binomial calculator I can walk you through.

http://stattrek.com/online-calculator/binomial.aspx

The probability on a single success is 1/6 or roughly .1666667.
The number of trials is your dice pool size.
The number of successes in your given example is 2.
The probability that you're looking for is "Cumulative Probability: P(X > 2)"

To find the probability for 3 or more successes, just change the number of successes to 3.


Edit: Added more information and clarified the Excel syntax.

No Excel, but that calculator seems easy to use! Thanks!

Here's what I got. I hope I did this right! I used the bottom Cumulative Probability though (greater than or equal rather than greater) - unless I'm misunderstanding something, that's the one I want? The difference is quite large.

2 sixes:
2d6 = 0.027 = 3%
3d6 = 0.07 = 7%
4d6 = 0.13 = 13%
5d6 = 0.196 = 20%
6d6 = 0.263 = 26%
7d6 = 0.33 = 33%
8d6 = 0.4 = 40%
9d6 = 0.457 = 46%
10d6 = 0.515 = 51%
11d6 = 0.57 = 57%
12d6 = 0.618 = 62%
13d6 = 0.66 = 66%
14d6 = 0.7 = 70%
15d6 = 0.74 = 74%
16d6 = 0.77 = 77%
17d6 = 0.8 = 80%
18d6 = 0.827 = 83%
19d6 = 0.85 = 85%
20d6 = 0.87 = 87%

3 sixes:
3d6 = 0.005 = 0.5%
4d6 = 0.016 = 1.6%
5d6 = 0.035 = 3.5%
6d6 = 0.06 = 6%
7d7 = 0.095 = 10%
8d6 = 0.13 = 13%
9d6 = 0.178 = 18%
10d6 = 0.225 = 23%
11d6 = 0.27 = 27%
12d6 = 0.32 = 32%
13d6 = 0.37 = 37%
14d6 = 0.42 = 42%
15d6 = 0.47 = 47%
16d6 = 0.51 = 51%
17d6 = 0.56 = 56%
18d6 = 0.597 = 60%
19d6 = 0.63 = 63%
20d6 = 0.67 = 67%

 

[Reinhart]

No Excel, but that calculator seems easy to use! Thanks!

Here's what I got. I hope I did this right! I used the bottom Cumulative Probability though (greater than or equal rather than greater) - unless I'm misunderstanding something, that's the one I want? The difference is quite large.

It looks correct after a few spot-checks. I did mean to use the the greater-than/less-than sign, but when I copy and pasted it into here I guess it didn't take.

 

[Desh-Rae-Halra]

I'm going to start generating stats by rolling 20d6 and dropping the lowest 17