A and B alternately throw a pair of dice. A wins if he throws a sum of 5 before B throws a sum of 8 , and B wins if he throws a sum of 8 before A throws a sum of 5 . The probability, that A wins if A

Question

A and B alternately throw a pair of dice. A wins if he throws a sum of 5 before B throws a sum of 8 , and B wins if he throws a sum of 8 before A throws a sum of 5 . The probability, that A wins if A makes the first throw, is

TestHub · Accepted Answer

For sum ' 5 ' → ( 1 , 4 ) , ( 2 , 3 ) , ( 3 , 2 ) ( 4 , 1 ) ⇒ P ( A ) = 36 4 ​ For sum ' 8 ' → ( 2 , 6 ) , ( 3 , 5 ) , ( 4 , 4 ) For sum ' 5 ' → ( 1 , 4 ) , ( 2 , 3 ) , ( 3 , 2 ) ( 4 , 1 ) ⇒ P ( A ) = 36 4 ​ For sum ' 8 ' → ( 2 , 6 ) , ( 3 , 5 ) , ( 4 , 4 ) ​ ( 5 , 3 ) , ( 6 , 2 ) ⇒ P ( B ) = 36 5 ​ P ( A ˉ ) = 36 32 ​ ​ , P ( B ˉ ) = 36 31 ​ ​ P ( A wins ) = = ​ P ( A ) + P ( A ˉ ) P ( B ˉ ) P ( A ) + + P ( A ˉ ) P ( B ˉ ) P ( A ˉ ) P ( B ˉ ) P ( A ) + … 1 − P ( A ˉ ) P ( B ˉ ) P ( A ) ​ = 19 9 ​ ​ ​

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