Randomized algorithms are playing a more and more important role in computation because of their simplicity and fastness. But sometimes the good performance of randomized algorithms does not require completely independent random variables as their input. In this paper, a new random algorithm is introduced for the classical problem of estimating the cardinality of a union of sets, which only needs pair-wise independent random input. This approach helps to reduce the random bits used in the algorithm. For fixed accuracy parameter ε and confidence parameter δ, the algorithm needs O(t^1/2) random bits, much fewer than those of a standard randomized algorithm O(tlogtM). And the running time bounds of the algorithm do not increase essentially O(t²logM), where t is the number of sets and M is the maximal cardinality of an individual set).