The overwhelming majority of positions using the full original set of 32 pieces are illegal and cannot arise in a legal game of chess. Specifically, the proportion of legal positions among all those using the 32 piece set is at most $4.0763\cdot 10^{-10}$.

To see this, consider a legal position using the 32 piece set. Since it has 32 pieces, there can have been no captures yet. In particular, each pawn must still be on its original file, not in the first or last rank, and furthermore, still opposed by the opposite-color pawn still facing it on that file. Within each file, therefore, you can easily count precisely 15 arrangements of one black pawn and one white pawn that exhibit this feature. Thus, there are precisely $15^8$ many ways to arrange the pawns overall in such a way that the position is not immediately seen as illegal. But there are are ${64 \choose 8}\cdot{56\choose 8}$ many ways to arrange the $16$ black and white pawns on the board. Since $${15^8 \over {64\choose 8}\cdot{56\choose 8}}= { 2562890625\over 4426165368\cdot 1420494075}\approx 4.0762706\cdot 10^{-10},$$ we get the upper bound as claimed.

The overwhelming majority of positions using the full original set of 32 pieces are illegal and cannot arise in a legal game of chess. Specifically, the proportion of legal positions among all those using the 32 piece set is strictly less than $4.0763\cdot 10^{-10}$.

To see this, consider a legal position using the 32 piece set. Since it has 32 pieces, there can have been no captures yet. In particular, each pawn must still be on its original file, not in the first or last rank, and furthermore, still opposed by the opposite-color pawn still facing it on that file. Within each file, therefore, you can easily count precisely 15 arrangements of one black pawn and one white pawn that exhibit this feature. Thus, there are precisely $15^8$ many ways to arrange the pawns overall in such a way that the position is not immediately seen as illegal. But there are are ${64 \choose 8}\cdot{56\choose 8}$ many ways to arrange the $16$ black and white pawns on an empty board. For each arrangement of the pawns, there are exactly the same number of ways to arrange the remaining pieces. The proportion of legal positions using the full 32 piece set is therefore at most $${15^8 \over {64\choose 8}\cdot{56\choose 8}}= { 2562890625\over 4426165368\cdot 1420494075}\approx 4.0762706\cdot 10^{-10},$$ and so we get the upper bound as claimed.