The generalized stacking fault energy (γ surface) provides fundamental physics for understanding the plastic deformation mechanisms. Using the ab initio exact muffin-tin orbitals method in combination with the coherent potential approximation, we calculate the γ surface for the disordered Cu-Al, Cu-Zn, Cu-Ga, Cu-Ni, Pd-Ag and Pd-Au alloys. Studying the effect of segregation of the solute to the stacking fault planes shows that only the local chemical composition affects the γ surface. The calculated alloying trends are discussed using the electronic band structure of the base and distorted alloys.Based on our γ surface results, we demonstrate that the previous revealed 'universal scaling law' between the intrinsic energy barriers (IEBs) is well obeyed in random solid solutions. This greatly simplifies the calculations of the twinning measure parameters or the critical twinning stress. Adopting two twinnability measure parameters derived from the IEBs, we find that in binary Cu alloys, Al, Zn and Ga increase the twinnability, while Ni decreases it. Aluminum and gallium yield similar effects on the twinnability.