dice roll n in N 1.r

sides <- 8
lim <- 6
n <- 20
lb <- 12 
ub <- 16

pbinom(ub, n, (sides-lim)/sides) - pbinom(lb-1, n, (sides-lim)/sides)
# 0.000935362

dice roll n in N 2.r

fc <- function(x) { # Factorial
    x[x == 0] <- 1
    vapply(x, function(y) prod(1:y), numeric(1))
}
ch <- function(n, k) fc(n)/(fc(k)*fc(n-k)) # Binomial coefficient
db <- function(x, n, p) ch(n, x) * p^x * (1-p)^(n-x) # PMF
pb <- function(q, n, p) sum(db(0:floor(q), n, p)) # CDF

pb(ub, n, (sides-lim)/sides) - pb(lb-1, n, (sides-lim)/sides)
# 0.000935362