A pair of unbiased dice is rolled together till a sum of either 5 or 7 is obtained. The probability that 5 comes before 7 is

Let A denote the event that a sum of 5 occurs, B the event that a sum of 7 occurs, and C the event that neither a sumof 5 nor a sum of 7 occurs. We have,P(A)=436=19,P(B)=636=16P(C)=2636=1318Thus, probability that A occurs before B isP[A or (C∩A) or (C∩C∩A) or …]=P(A)+P(C∩A)+P(C∩C∩A)+⋯=P(A)+P(C)P(A)+P(C)2P(A)+⋯=19+1318×19+1318219+⋯=1/91−13/18=25