Let C _ 1 and C _ 2 be two biased coins such that the probabilities...

For tossing two coins independently, maximum and minimum heads that can appear will be $2 \text{ and } 0$ respectively.

Roots of equation ${x}^{2} - \alphax + \beta = 0$ are real and equal, then discriminant $D = 0$

$⇒ {\alpha}^{2} - 4\beta = 0 ⇒ {\alpha}^{2} = 4\beta$

$⇒ \left(\alpha = 0, \beta = 0\right) or \left(\alpha = 2, \beta = 1\right)$

Thus, the required probability

$= C0 2 {\left(\dfrac{2}{3}\right)}^{0}{\left(\dfrac{1}{3}\right)}^{2} . C0 2 {\left(\dfrac{1}{3}\right)}^{0}{\left(\dfrac{2}{3}\right)}^{2} + C2 2 {\left(\dfrac{2}{3}\right)}^{2}{\left(\dfrac{1}{3}\right)}^{0} . C1 2 {\left(\dfrac{1}{3}\right)}^{1}{\left(\dfrac{2}{3}\right)}^{1}$

$= \dfrac{1}{9} \times \dfrac{4}{9} + \dfrac{4}{9} \times \dfrac{4}{9} = \dfrac{20}{81}$