What a game!

$First,\quad let\quad A1\quad denote\quad that\quad A\quad wins\quad from\quad first\quad throw,B1\quad denotes\quad B\quad wins\quad from\quad first\quad throw\\ \quad \quad \quad \quad \quad let\quad A2....\quad from\quad second\quad throw,B2.....from\quad second\quad throw,\quad and\quad so\quad on.....\\ we\quad know\quad that\quad P(A1)=\frac { 1 }{ 6 } ,P(\overline { A1 } )=\frac { 5 }{ 6 } ,P(B1)=\frac { 5 }{ 6 } \times \frac { 1 }{ 6 } ,P(A2)={ (\frac { 5 }{ 6 } ) }^{ 2 }\times \frac { 1 }{ 6 } \quad ...(you\quad can\quad draw\quad a\quad tree\quad diagram)\\ so,\quad P(A\quad wins)=P(A1)+P(\overline { A1 } \overline { B1 } A2)+P(\overline { A1 } \overline { B1 } \overline { A2 } \overline { B2 } A3)+.........\\ \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad =\quad \frac { 1 }{ 6 } +({ (\frac { 5 }{ 6 } ) }^{ 2 }\times \frac { 1 }{ 6 } )\quad +({ (\frac { 5 }{ 6 } ) }^{ 4 }\times \frac { 1 }{ 6 } )+({ (\frac { 5 }{ 6 } ) }^{ 6 }\times \frac { 1 }{ 6 } )+.......\\ \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad =\frac { 1 }{ 6 } \times \frac { 1 }{ 1-{ (\frac { 5 }{ 6 } ) }^{ 2 } } =0.54\quad (it's\quad a\quad geometric\quad sequence\quad of\quad infinite\quad terms)\\ \\ \\ \\$