An S=1/2 ferromagnetic (F) and antiferromagnetic (AF) random alternating Heisenberg quantum spin chain model is investigated in connection to its realization compound (CH3)(2)CHNH3Cu(ClxBr1-x)(3). The exchange interaction bonds consist of alternating strong F-AF random bonds and weak uniform AF bonds. Using the quantum Monte Carlo method we have found that the excitation energy gap closes and the uniform AF order becomes critical in the intermediate-concentration region. This finding explains the experimental observation of the magnetic transition by considering weak interchain interactions. The present results suggest that uniform AF order survives even in the presence of randomly located ferromagnetic bonds. This may be an exotic quantum effect.