The problem of drawing a person from a household often occurs at the final stage of a health survey design. In case of such designs, selection probabilities pertaining to the households are usually more or less equal. Sampling procedures that are used to select a person within households intend to be also quasi-random, i.e. conditional selection probabilities pertaining to persons living in the same household are equal. We found that, contrary to the widely held opinion, the above design is not capable of providing representativity by gender and age. We suggest a modification of the Kish grid, which is one of the widely used quasi-random procedures. The modified grid is more appropriate for selecting a representative sample. Since the performance of the grid depends on the household structure within the target population, its modification varies country by country. Those countries are considered in which the Kish grid is regularly used in surveys.