001/ Challenge problem http://mathhelpboards.com/challenge-questions-puzzles-28/dice-rolling-running-total-probability-23339.html#post104363

A standard six–sided dice is rolled repeatedly and a running total is kept of
all the numbers rolled. Which of 2,6,1006 is more likely to be one of these
totals? Prove your answer.

We http://mathhelpboards.com/challenge-questions-puzzles-28/dice-rolling-running-total-probability-23339.html#post104363

total number of rolls maximum considered shall be 1006.( Because 1007 counts shall give a minumum sum 1007
let us count number of ways partial sums can be
2 = 2 or 1 + 1 so 2 followed by any number or 1 ,1 followed by any number can give sum 2 so 

$2 *6^{1005} + 1 * 1 * 6^{1004}$ ways
6 csn come as (6), (5,1), (4,2)  and other compinations so number of ways 6 can come
$> 6^{1005} + 2 * 6^{1004}$ taking care if above 3 cases and there are more cases also but not requires

for 1 to come we can have all 1 and for $1006^{th}$ element to be 1 the number of ways is $6^{1005}$ out of which some and lot all can have partial sum 1006.

comparing above 3 we see that  number of ways partial sum > 7 is largest. then comes number of ways partial sum 2 then 1006.
so 6 is most likely to occur followed by 2 followed by 1006.