During the character creation process in Dungeons & Dragons (D&D), if one chooses to roll for their attributes (dexterity, charisma, constitution, wisdom, intelligence, strength), they must roll 4 6-sided dice 6 times, each time taking the total of the top three dice with the highest outcomes. This is a simulation study (N=10,000) using R of dis…

Distributions.md

Dice Combination	Proportion
14, 13, 12, 11, 10, 9	0.0015
16, 15, 14, 13, 12, 11	0.0015
15, 14, 13, 12, 10, 9	0.0014
16, 15, 14, 13, 11, 10	0.0014
14, 13, 13, 12, 11, 10	0.0013
15, 14, 13, 12, 12, 11	0.0013
13, 13, 12, 11, 11, 10	0.0012
14, 14, 13, 13, 12, 11	0.0012
15, 14, 13, 12, 11, 10	0.0012
15, 14, 13, 12, 11, 9	0.0012
16, 13, 12, 11, 10, 9	0.0012
16, 14, 13, 12, 11, 10	0.0012
14, 13, 12, 12, 9, 8	0.0011
15, 13, 12, 12, 11, 10	0.0011
15, 14, 13, 12, 11, 7	0.0011
15, 14, 14, 13, 12, 11	0.0011
16, 14, 13, 12, 11, 8	0.0011
16, 14, 13, 12, 11, 9	0.0011
16, 15, 13, 13, 12, 11	0.0011
14, 13, 12, 11, 10, 8	0.0010

source.R

library(magrittr) # install.packages('magrittr')
library(knitr) # install.packages('knitr')
n.sims <- 1e4
set.seed(0)
distrs <- replicate(n.sims,
		    replicate(6,sample(1:6,4,repl=T) %>% { sort(.,decreasing=T)[1:3] } %>% sum )) %>%
		matrix(ncol=6,nrow=n.sims,byrow=T) %>% apply(1,. %>% sort(decreasing=T)) %>% t

# Plot
distrs %>% as.vector %>% { table(.)/(n.sims*6) } %>%
	{ x <- barplot(.,main="Distribution of Attribute Rolls",xlab="Total of Top Three Dice",ylab="Proportion",col="cornflowerblue",border="white",ylim=c(0,0.15));
		cbind(x,.) } %>% { text(.[,1],.[,2],labels=round(.[,2],3),pos=3,cex=0.75,col="cornflowerblue") }

# Table
apply(distrs,1,paste,collapse=", ") %>%
	{ table(.)/n.sims } %>% sort(decreasing=T) %>% head(20) %>%
		as.data.frame %>% { names(.) <- 'Proportion'; . } %>% kable