Is it permissible to pray on the carpets of government institutions, knowing that people walk on them with their shoes and it is not known whether there is any impurity on them or not?

Among the well-known rules is that the default state of things is purity until their impurity is ascertained. Although shoes are predominantly impure, the Sacred Law has overridden the حكم الغالب (ruling based on predominance) and established the حكم النادر (ruling based on rarity), as the Sunnah permitted prayer in shoes out of mercy and ease for the servants. Therefore, there is no harm in praying on the aforementioned carpet unless its impurity is confirmed.