[MickeyHoldem]

Here's some raw data:

Rolling 5 dice... 7776 outcomes
No 6's......3125/7776 - 40.19%
6 no 5's....2101/7776 - 27.02%
65 no 4's...1320/7776 - 16.98%
654 qual....1230/7776 - 15.82%

654 qual with score 11,12......50/7776 - 0.64%

Roling 4 dice... 1296 outcomes
No 5's......625/1296 - 48.23%
5 no 4's....369/1296 - 28.47%
54 qual.....302/1296 - 23.30%

54 qual w/ score 11,12.........24/1296 - 1.85%

Rolling 3 dice... 216 outcomes
no 4........125/216 - 57.87%
4 qual.......91/216 - 42.13%
4 qual w/ score 11,12..........12/216 - 5.56%


Here's the calculation:

= 50/7776
+ 1320/7776 * (9/216+(125/216*(9/216+(125/216*9/216))))
+ 2101/7776 * (24/1296+625/1296*(24/1296+(625/1296*24/1296)+(369/1296*9/216))+369/1296*(9/216+(125/216*9/216)))
+ 3125/7776*(50/7776+(1320/7776*(9/216+(125/216*9/216)))+2101/7776*(24/1296+(625/1296*24/1296)+369/1296*9/216))+3125/7776*(50/7776+(1320/7776*9/216)+(2101/7776*24/1296)+(3125/7776*50/7776))

Here's the calculation to just roll a qualifier:

=1230/7776
+ 1320/7776*(91/216+(125/216*(91/216+(125/216*91/216))))
+ 2101/7776*(302/1296+625/1296*(302/1296+(625/1296*302/1296)+(369/1296*91/216))+369/1296*(91/216+(125/216*91/216)))
+ 3125/7776*(1230/7776+(1320/7776*(91/216+(125/216*91/216)))+2101/7776*(302/1296+(625/1296*302/1296)+369/1296*91/216))+3125/7776*(1230/7776+(1320/7776*91/216)+(2101/7776*302/1296)+(3125/7776*1230/7776))