How do we know that change in entropy(dQ/T) & change in randomness are the same thing? It makes sense to define a state function dQ/T such that its integral over a cycle is zero. But how can we correlate it with randomness in such simple terms?

entropy is nothing but disordensess of a thing.disorderness is measured by randomness.for ex(take two cylinders of oxygen and hydrogen.a molecule different from these two gases entering into a cylinder has the probability of 1/2.for two molecules it is (1/2)^2.means here entropy is 1/2 for a molecule.for n no. of molecules  it is pln(w)).here a body or a non-living thing lossing its information in the form of heat in engines.means its disorderness is increasing. the heat loosing for a particular temp will give the entropy.for living things entropy will decrease because we wiil get  orderness with time .