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Let 퐺 be a finite group, and let 퐻 be a subgroup of 퐺. We compute the probability, denoted by P G ⁢ (H) , that a left transversal of 퐻 in 퐺 is also a right transversal, thus a two-sided one. Moreover, we define, and denote by tp ⁡ (G) , the common transversal probability of 퐺 to be the minimum, taken over all subgroups 퐻 of 퐺, of P G ⁢ (H) . We prove a number of results regarding the invariant tp ⁡ (G) , like lower and upper bounds, and possible values it can attain. We also show that tp ⁡ (G) determines structural properties of 퐺. Finally, several open problems are formulated and discussed. [ABSTRACT FROM AUTHOR]