Simulating the 4 Aces

Deal a standard deck out into a sequence of all 52 cards. What is the probability that all 4 aces are adjacent to one another?

We know the analytical approach give the answer:

49/choose(52, 4)

## [1] 0.0001809955

Method 1

Deal all cards out, then look at aces for adjacency.

n <- 1e5
simlist <- rep(NA, n)

# replicate
system.time(
for(i in 1:n) {
  # simulate
  deal <- sample(1:52)
  ace_indices <- which(deal %in% 1:4)
  # success?
  simlist[i] <- if(max(diff(ace_indices)) == 1) 1 else 0
}
)

##    user  system elapsed 
##   2.110   0.043   2.157

# estimate
mean(simlist)

## [1] 0.00018

Method 2

Select the indices of the aces directly, then check for adjacency.

n <- 1e5
simlist <- rep(NA, n)

# replicate
system.time(
for(i in 1:n) {
  # simulate
  ace_indices <- sample(1:52, size = 4)
  # success?
  simlist[i] <- if(max(diff(sort(ace_indices))) == 1) 1 else 0
}
)

##    user  system elapsed 
##   4.576   0.025   4.604

# estimate
mean(simlist)

## [1] 0.00017