1.  apply()

2.  Let R be the initial roll, and B_1 and B_2 be bonus rolls, if any,
from squares 1 and 2 respectively.

P(not yet complete circuit) 
   = 1 - P(complete circuit)
   = 1 - P(R = 1 and B_1 = 1 and B_2 = 6 or
           R = 2 and B_2 = 6)
   = 1 - (1/6) (1/6) (1/6) - (1/6) (1/6)

3.

nreps <- 10000
nstops <- 2  # note 2, not 10
countCondition <- 0
countOnly1 <- 0
for (i in 1:nreps) {
   passengers <- 0
   for (j in 1:nstops) {
      oldpass <- passengers  # number arrived at this stop
      if (passengers > 0) 
         for (k in 1:passengers) 
            if (runif(1) < 0.2) 
               passengers <- passengers - 1
      newpass <- sample(0:2,1,prob=c(0.5,0.4,0.1))
      passengers <- passengers + newpass  # number leave this stop
   }       
   if (passengers == 0 && oldpass > 0) {
      countCondition <- countCondition + 1
      if (oldpass == 1) countOnly1 <- countOnly1 + 1
   }                                 
}                                    
print(countOnly1/countCondition)

4.

simAttach <- function(adjMat)    
{                      
   k <- nrow(adjMat) + 1
   degrees <- apply(adjMat,1,sum)
   prbs <- degrees / sum(degrees)
   return(sample(1:(k-1),1,prob=prbs))
}
                         
# test it                
testAM <- rbind(c(0,1,0,0),c(1,0,1,0),c(0,1,0,1),c(0,0,1,0))
reps <- replicate(10000,simAttach(testAM))
table(reps) / 10000