Take care that not all of king and pawn versus king are drawn. Their result depends on relative position of kings.

Nice example of more extreme imbalance is that sometimes King+Queen vs King+Pawn is drawn - that is when pawn is on seventh rank of bishop or rook file with defending king in front of it, while attacking king is not close enough to force checkmate.

See: https://en.wikipedia.org/wiki/Queen_versus_pawn_endgame#Quee...