Proposed an one-dimensional sandpile model which include avalanche probability, and performed computer simulation by cellular automata method. The results show that there are two critical points p1 and p2 when avalanche probability p transits from 0 to 1. The self-organized criticality(SOC) behavior can be found in the model when p12. There is a sharp transition between the trivial behaviour and the SOC behaviour in the model. When there is SOC, the SOC behaviour is universal, the two critical exponents are 1.50±0.02 and 1.58±0.15. With the model, the SOC phenomenon appearing in the experiment of one-demensional rice-pile is well explained.