The specification of in these sampling techniques are as follows. A simple choice of for all state will lead the estimator

As the estimator leads to

the ensemble averaged value of the property A. This is known as simple sampling MC method.

As already mentioned, at low temperature the system remains restricted in an extremely restricted part of the phase space and an importance sampling is required. The Mc method should automatically lead us to the important region of phase space. One could sample points preferentially from the region which is populated by the appropriate states. This is realized in the following manner: Instead of picking N states randomly with equal probability, pick them with a probability , the correct Boltzmann weight. Then the expectation value of a quantity will be given by

which simply reduces to

where each state s is picked with the correct Boltzmann probability (not randomly selected states with equal probability as in simple sampling). This averaging is called `importance sampling' average. Hoverer, it is not a easy task to pick up states which appear with its correct Boltzmann weight. It is possible to realize importance sampling with the help of Markov chain.