I worked it out for P2, P3, P4, and quinto. "Normal" is the probability under normal drawing condtions and "weird" is the probability under the puzzle's conditions.
Pick 2
Doubles: Normal = 10%, Weird = 18.2%, Change ?
Singles: Normal = 90%, Weird = 81.8%, Change ?
Pick 3
Triples: Normal = 1%, Weird = 3.31%, Change ?
Doubles: Normal = 27%, Weird = 37.2%, Change ?
Singles: Normal = 72%, Weird = 59.5%, Change ?
Pick 4
Quads: Normal = 0.1%, Weird = 0.601%, Change ?
Triples: Normal = 3.6%, Weird = 8.11%, Change ?
Double-Doubles: Normal = 2.7%, Weird = 4.73%, Change ?
Doubles: Normal = 43.2%, Weird = 48.7%, Change ?
Singles: Normal = 50.4%, Weird = 37.9%, Change ?
Quinto
Quints: Normal = 0.01%, Weird = 0.109%, Change ?
Quads: Normal = 0.45%, Weird = 1.72%, Change ?
Triple-Doubles: Normal = 0.9%, Weird = 2.27%, Change ?
Triples: Normal = 7.2%, Weird = 12.3%, Change ?
Double-Doubles: Normal = 10.8%, Weird = 14.75%, Change ?
Doubles: Normal = 50.4%, Weird = 48.2%, Change ?
Singles: Normal = 30.24%, Weird = 20.65%, Change ?
It makes sense intuitively that the weird drawing procedure favors combos with more repeated digits. Obviously the smart bet in Pick 3 would be to play triples, assuming the payout for a straight is the same no matter what numbers are drawn.
EDIT: The question marks are supposed to be up and down arrows that aren't showing up properly.