Using R for dice probabilities Using R, what function(s) would I use to obtain the following probabilities? 


*

*Roll at least one 1 when rolling 2 six-sided dice (2d6) = 11/36

*Roll at least one 1 when rolling 3 six-sided dice (3d6) = 91/216

*Roll at least one 1 when rolling 1d4, 1d6, 1d8, and 1d8 = 801/1536


First I hope my answers above are correct!  I did these pretty much manually.
I think I need to use binomial distributions and/or probability-generating functions, but not sure if I'm over-complicating things.  I've tried using R's *binom() functions but can't seem to arrive at the answers I need.
 A: Solution 1
There is a package dice that could help you. I got the same probabilities for the first two examples, but not for the last one. See the example code:
require(dice)

# Roll at least one 1 when rolling 2 six-sided dice (2d6) = 11/36
getEventProb(nrolls = 2, ndicePerRoll = 1, nsidesPerDie = 6,
             eventList = list(1))
11/36

# Roll at least one 1 when rolling 3 six-sided dice (3d6) = 91/216
getEventProb(nrolls = 3, ndicePerRoll = 1, nsidesPerDie = 6,
             eventList = list(1))
91/216

# Roll at least one 1 when rolling 1d4, 1d6, 1d8, and 1d8 = 654/1536
p1 <- getEventProb(nrolls = 1, ndicePerRoll = 1, nsidesPerDie = 4,
                   eventList = list(1))
p2 <- getEventProb(nrolls = 1, ndicePerRoll = 1, nsidesPerDie = 6,
                   eventList = list(1))
p3 <- getEventProb(nrolls = 2, ndicePerRoll = 1, nsidesPerDie = 8,
                   eventList = list(1))

p1 + p2 + p3 - p1 * p2 - p2 * p3 - p3 * p1 + p1 * p2 * p3
801/1536

### The same as
1 - (1-p1) * (1-p2) * (1-p3)
801/1536

Solution 2
This is a solution with out usage of any package. You can compute the probability to draw at least one $1$ by this formula (mentioned by @whuber):
$$ p = 1 - \prod_{i=1}^n \left( 1 - \frac{1}{d_i} \right) $$
where $n$ is the number of dices and $d_i$ is the number of sides of dice $i$.
Then you can define a function in R with one argument dices, where dices is a vector of sides. See the example code:
dice.prob <- function(dices) 1 - prod(1 - 1 / dices)

dice.prob(c(6, 6))
11/36

dice.prob(c(6, 6, 6))
91/216

dice.prob(c(4, 6, 8, 8))
801/1536

