Probability USC Problem Error? A person rolls four fair six-sided die. What is the probability that the person
rolls exactly one 1 and exactly one 2?
This is rather a simple probability problem, but I scored incorrectly for some reason. I first discerned how many total possibilities there were which was $\frac{1}{6^4}$. I then figured out how many checked out. I did $\frac{(1*1*4*4)(4!)}{6^4}$; one way to choose 1, one way to choose 2, and since there can only be exactly one of 1 and 2, the other two die must be 4. I then multiplied by 4! for 24 different arrangements. I resulted in an answer of $\frac{8}{27}$ but the answer is $\frac{4}{27}$. 
Help is appreciated. 
 A: Multinomial probability. Consider three kinds of outcomes in each of $n=4$ trials:  $1$'s, $2$'s, and
'something else', with probabilities $(1/6), (1.6),$ and $(4/6),$ respectively.
$$P(\text{One 1 and One 2}) = {4 \choose {1,1,2}}(1/6)^1(1/6)^1(4/6)^2
= \frac{4!}{1!\cdot 1!\cdot 2!}\frac{4^2}{6^4} = \frac{4}{27} = 0.1481481.$$
Simulation. In R statistical software we simulate the 4-roll experiment
a million times. At the end, the vector event has a million TRUEs and FALSEs; the mean of such a logical vector is its proportion of TRUEs.
With a million iterations, the result 0.148 should be accurate to at least two decimal places (in this particular case, three places).
set.seed(625)  # for exact same simulation, retain this; omit for fresh simulation
m = 10^6; event = logical(m)
for(i in 1:m){
  s = sample(1:6, 4, repl=T)
  event[i] = (sum(s==1)==1) & (sum(s==2)==1) 
  }
mean(event)
## 0.147672           # aprx probability 'One 1 and One 2'
2*sd(event)/sqrt(m)
## 0.0007095495       # 95% margin of simulation error

Note: Second, more efficient, logically equivalent simulation. It uses
pseudorandom numbers in exactly the same way as did the first simulation, so it gets
exactly the same result starting with the same seed.
set.seed(625) 
m = 10^6;  die = c(1, 10, 0, 0, 0, 0)  # 1=1, 10=2, 0=Else
s = replicate( m, sum(sample(die, 4, repl=T)) )
mean(s==11)
## 0.147672

