Sum of roll of N dice

[Dieter]

(05-02-2018 05:06 AM)Joe Horn Wrote:  THE QUESTION: Is there a direct way to return the sum of a roll of N dice without generating N random numbers and adding them together?

Here is another possible method – but I have no idea if it makes sense here. So please correct me if I'm wrong.

Let's assume you roll a really BIG number of dice. Not just 3 or 5, maybe n=10 or 50. According to the central limit theorem this should approximate a Normal distribution. If we assume tossing a single die does not only result in six disctinct outcomes (1, 2, 3, 4, 5, 6) but a continuous result (1...6), the mean is (1+6)/2 = 3,5 and the variance is n · (6–1)²/12. Now simply generate a normally distributed random number, i.e. apply the Normal quantile function (or an approximation thereof) to the usual RAN# result, then rescale it with mean + √var · ran#.

I have tried this on the WP34s and it seems to work fine, even for n as low as 5. Maybe the two parameters require an adjustment (is it 6–1 or 6 in the variance?), but... is the general idea a possible solution, at least for larger n ?

For the record, here is a result for 1000 tosses of 4 dice:

Code:

sum  frequency (w/mod. σ)
-------------------------
4       0         0
5       0         6
6       2        10
7       9        14
8       18       19
9       23       47
10      63       60
11      77       73
12      100      94
13      121     100
14      147     125
15      116      98
16      109      99
17      87       84
18      58       64
19      34       43
20      16       26
21      13       18
22      2         8
23      3         6
24      3         7

Note: the right column shows the frequencies based on the modified variance, i.e. σ² = n · 6²/12.

Edit: for a discrete random variable (1, 2, 3, 4, 5, 6) the variance seems to be 35/12, so I tried another run with σ² = n · 35/12. The results were similar and looked good either.

Dieter