In an infinite set, probabilities are defined on the structure of the set rather than on individual elements. We should take into account the property of a σ-algebra where probabilities are defined. A σ-algebra is closed under ‘only countable’ unions, and the axioms of probability assume σ-additivity. If this is overlooked, something bizarre could be happened as the proposed three solutions of Bertrand's problem. Bertrand's problem is not a paradox, but well defined(posed). The suggested three solutions have the common problem of dividing the sample space into an uncountably infinite number of sets and treating them equally. If a set is divided into equal(treated) and uncountable infinity, all the divided sets have probability 0, so calculating conditional probabilities with these sets or comparing them with each other becomes meaningless. In a sample space composed of an uncountably infinite number of elements such as [0,1], after calculating the number of cases using the sets(J-sequence m-collection cover) generated by equally dividing the sample space into finite numbers, the probability of an event can be calculated with its limit value(as m becomes infinite). The answer of Bertrand's problem is 1/3.