Rolling a fair dice $7$ times, what is the probability that the total sum is divisible by $3$, and no odd value was rolled?
I know that there are $3^6$ possible ways for the first $6$ dice to give only even values, but how many of them are divisible by $3$ ?
Best Answer
If we forget about this "no-odd" thing for a second, and just focus on "divisible by $3$", imagine having thrown the die six times already, and just waiting for the seventh roll to decide everything. How many different results on the seventh die will make the total result divisible by $3$? Does it matter what the sum of the first $6$ are?
Now adapt this to the case where you're only throwing evens. Does that really change anything of the above argument?