]> $$I_x^{-1}\left(a,b\right)={\left(apB\left(a,b\right)\right)}^{\frac{1}{a}}+\frac{b-1}{a+1}{\left(apB\left(a,b\right)\right)}^{\frac{2}{a}}+\frac{\left(b-1\right)\left(a^2+3ba-a+5b-4\right)}{2{\left(a+1\right)}^2\left(a+2\right)}{\left(apB\left(a,b\right)\right)}^{\frac{3}{a}}$$ $$I_x\left(a,a+\beta \right);\beta <\sqrt{a}$$ $$I_x\left(a,b\right);0.2\le \frac{a}{a+b}\le 0.8$$ $$I_x^{-1}\left(a,b\right)={\left(apB\left(a,b\right)\right)}^{\frac{1}{a}}$$ $$p=1-q=I_x\left(a,b\right)=1-I_y\left(b,a\right);y=1-x$$ $$y=1-I_x^{-1}\left(a,b\right)={\left(bqB\left(a,b\right)\right)}^{\frac{1}{b}}$$ $$x_0=\frac{x_g}{1+x_g};x_g={\left(apB\left(a,b\right)\right)}^{\frac{1}{a}}$$ $$y={\left(1-p^{bB\left(a,b\right)}\right)}^{\frac{1}{b}}$$