A quadratic equation is chosen from the set of all quadratic equations which are unchanged by squaring their roots. The chance that the chosen equation has equal roots, is

Let $\alpha$ and $\beta$ be the roots of the quadratic equation. According to question,

$\alpha +\beta ={\alpha }^2+{\beta }^2\text{ and }\alpha \beta ={\alpha }^2{\beta }^2\Rightarrow \alpha \beta (\alpha \beta -1)=0$

$\Rightarrow \alpha \beta =1$ or $\alpha \beta =0$

$\Rightarrow \alpha =1,\beta =1;\alpha =\omega \beta ={\omega }^2$ [cube roots and unity] 

$\alpha =1, \beta =0; \alpha =0, \beta =0$

$\therefore n\left(S\right)=$ Number of quadratic equations which are unchanged by squaring their roots $=4$

and $n\left(E\right)=$ Number of quadratic equations have equal roots

$=2$

$\therefore$Required probability $=\frac{n\left(E\right)}{n\left(S\right)}=\frac{2}{4}=\frac{1}{2}$