Conditional probability appears to be a simple rule, but it is often misunderstood. Most researches examine students’ misconceptions  of conditional probability, but it is useless to estimate and blame students’ inability to understand this concept without analyzing the mathematical structure of conditional probability in various situations. This paper attempts to analyze conditional probability mathematically by two approaches to solving problems: “relative-conditional probability,” which finds the answer through the relative frequency ratio,   and “assumptive-conditional probability,” which finds the answer through inferring changed situations. This paper focuses on the difference between the two kinds of conditional probabilities, why they are applied differently in different situations, and how they are consistent in mathematical structure. In conclusion, the two kinds of conditional probabilities that appear different outwardly really have the same structure because Kolmogorov’s new definition of sample space permits two kinds of conditional probabilities with different situations and meanings to    have the same mathematical structure, inevitably creating students’ misunderstanding of conditional probability.

conditional probability, relative-conditional probability, assumptive-conditional probability, reduced sample space, misconception.