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Quantization for a Borel probability measure refers to the idea of estimating a given probability by a discrete probability with support containing a finite number of elements. If in the quantization some of the elements in the finite support are preselected, then the quantization is called a conditional quantization. In this paper, we have determined the conditional quantization, first for two different finite discrete distributions with a same conditional set, and for a finite discrete distribution with two different conditional sets. Next, we have determined the conditional and unconditional quantization for an infinite discrete distribution with support 12n:n∈N. We have also investigated the conditional quantization for an infinite discrete distribution with support 1n:n∈N. At the end of the paper, we have given a conjecture and discussed about some open problems based on the conjecture.