The evaluation of the probability of union of a large number of independent events requires several combinations involving the factorial and the use of high performance computers with several hours of processing. Bounds and simplifications on the probability of the union are useful in the analysis of stochastic problems across various areas including (but not limited to) systems reliability, biological systems, real-time fault-tolerant systems, probability theory, information theory and communications. We propose an approximation to evaluate the probability of the union of several independent events that uses the arithmetic mean of the probability of all of them. The approximate results are very close to, but larger than the exact values. The method allows a much smaller number of operations with a similar result and more simplicity.