statistics by dropping dice rolls..

okay, so, we all play D&D right? if not, quick explanation of terms:
a "d6" is a six-sided (cube) dice, which (unsurprisingly) contains the numbers 1, 2, 3, 4, 5 and 6 arranged with one value per face.

the term "1d6" means "roll 1 six-sided die" (1d6 has a sample space of {1...6}. "2d6" means "roll 2 six-sided dice and add their values together" sample space: {2...12}

the average of 1d6 is equal to (1+6)/2 = 7/2 = 3.5
the average of 3d6 is equal to 3*3.5 = 10.5

historically, character stats in D&D are generated via rolling 3d6. when 3rd edition was released oh-so-many years ago, they replaced the standard "3d6" generation method with "4d6, drop lowest roll"

Before we tackle the main event, let's take a simpler problem: 2d6 drop lowest.

for 2d6 drop lowest, the average is approximately 4.47. We can find this out by summing all 36 possible outcomes and then dividing by the number of possible outcomes. it's dirty and ugly, but trivial for 2 dice. 3 dice has a lot more combinations (6*6*6=216) :(

so, for 3d6, 4d6 and above, is there a more elegant solution?

Let's say the lowest number is dropped from 4 rolls. If this lowest number is one, then we have:

1***,

where the * can be 1 through 6.
If there are 0 1s amongst the ***, then there are (4,3)*5^3 choices ((4,1) denotes the binomial coefficient of 4 choose 1).
Likewise for j=1,2,3 1s amongst the ***, there are (4,3-j)*5^(3-j). So, there are sum((4,j)*5^j,j=0..3) ways to roll 4 6-sided dice where the 1 is dropped.

Generalizing, there are sum((n,j)*(s-k)^j,j=0..n-1) ways to roll n s-sided dice where k is dropped.

So, the probability of dropping k from n rolls of s-sided dice is found by P(k)=1/s^n*sum((n,j)*(s-k)^j,j=0..n-1).

Averaging these gives sum(k*P(k),k=1..s).

Thus the average value rolled after rolling n s-sided dice and dropping the lowest value should be

n*(s+1)/2-1/s^n*sum(sum(k*(n,j)*(s-k)^j,j=0..n-1),k=1..s),

which gives values for 6-sided dice:

2 dice -> avg. = 4.64
3 dice -> avg. = 8.49
4 dice -> avg. = 12.25,
etc.