Two dice are thrown Hence n(S) = 36   n(8) =5   n(7) =6

Let probability of A wining be P(W₁) = 5/36   P(L₁) =31/36

Let probability of B wining be P(W₂) = 6/36   P(L₂) =30/36

 

If A wins in I throw = 5/36

                II throw = 5/36 + 31/36 *30/36* 5/36

               III throw =  5/36 + 31/36 *30/36* 5/36 + (31/36)² *(30/36^{)2 *} 5/36

and so on ...........

 

Hence P(A) = 5/36 + 31/36 *30/36* 5/36 + (31/36)² *(30/36^{)2 *} 5/36 +............

                  =5/36 * 1/(1-31/36 *30/36)  = 30/61

 

If B wins in I throw = 31/36 *6/36

                II throw = 31/36 *6/36 + (31/36)² *30/36* 6/36

           III throw =  31/36 *6/36 + (31/36)² *30/36* 6/36+ (31/36)³ *(30/36)^{2 *} 6/36

and so on ...........

 

Hence P(B) = 31/36 *6/36 + (31/36)² *30/36* 6/36+ (31/36)³ *(30/36)^{2 *} 6/36+....

                  =31/36 *6/36 *1/(1-31/36 *30/36)  = 31/61

 

 

Hence P(A) : P(B) = 30 : 31