In order to balance the preference of the artificial entities towards exploration or exploitation (in their transition rule), a novel technique is proposed for replacing the random function used by the classical Ant Colony Optimization (ACO) algorithms for solving the Traveling Salesman Problem (TSP). The proposed Beta Distribution function (B), or random.betavariate(a, b) has the proven capability (depicted through test-runs) of influencing the algorithm's solution quality and convergence speed. Consequently, this paper will introduce in the related work section the classical ACO algorithm, with a focus on the transition rule used for choosing the next node in the problem's associated graph, followed by the related work on this topic, and it will continue with the introduction of the B function which will be presented both from a theoretical and practical perspective in relation with the scope: balancing between exploration and exploitation in order to improve the performance of the ACO algorithm for the TSP. The paper concludes that the B-EAS has the ability to find better solution than EAS for a set of benchmarks from the TSPLib library.