Let X(t) be a temporally homogeneous irreducible Markov chain in continuous time defined on 𝒩={0,1,...}. For k<i<j, let H={k+1,...,j-1} and let kTij(jTik) be the upward (downward) conditional first-passage time of X(t) from i to j(k) given no visit to 𝒩-H. These conditional passage times are studied through first-passage times of a modified chain HX(t) constructed by making the set 𝒩-H of states absorbing. It will be shown that the densities of kTij and jTik for any birth-death process are unimodal and the modes kmij (jmik) of the unimodal densities are non-increasing (non-decreasing) with respect to i. Some distribution properties of kTij and jTik for a time-reversible Markov chain are presented. Symmetry among kTij, jTik, kŢij and jŢik is also discussed, where k&Ttilde;ij and áMarkov chain are presented. Symmetry among
