Question

A and B throw two dice simultaneously turn by turn. A will win if he throws a total of 5, B will win if he throws a doublet. Find the probability that B will win the game, though A started the game.

Solution

∴ total number of possible outcomes = 36
Let E be the event of getting "a total of 5" in a throw of two dice.

therefore space space space space straight E space equals space left curly bracket left parenthesis 1 comma space 4 right parenthesis comma space left parenthesis 2 comma space 3 right parenthesis comma space left parenthesis 3 comma space 2 right parenthesis comma space left parenthesis 4 comma space 1 right parenthesis right curly bracket therefore space space space space straight P left parenthesis straight E right parenthesis space equals space 4 over 36 space equals space 1 over 9 comma space space space straight P left parenthesis straight E with bar on top right parenthesis space equals space 1 minus 1 over 9 space equals space 8 over 9

Let F be the event of getting a doublet in throw of two dice.

therefore

F = {(1, 1), (2, 2), (3, 3), (4, 4), (5, 5), (6, 6)}

therefore space space space straight P left parenthesis straight F right parenthesis space equals space 6 over 36 space equals space 1 over 6 comma space space straight P left parenthesis straight F with bar on top right parenthesis space equals space 1 minus 1 over 6 space equals 5 over 6

Now A starts the game.

therefore

B wins in second throw or fourth throw or 6th throw or ..........

therefore

probability of winning of B =

straight P left parenthesis straight E with bar on top right parenthesis space space straight P left parenthesis straight F right parenthesis space plus space straight P left parenthesis straight E with bar on top right parenthesis thin space straight P left parenthesis straight F with bar on top right parenthesis thin space straight P left parenthesis straight E with bar on top right parenthesis space straight P left parenthesis straight F right parenthesis

plus space straight P left parenthesis straight E with bar on top right parenthesis thin space straight P left parenthesis straight F with bar on top right parenthesis thin space straight P left parenthesis straight E with bar on top right parenthesis space straight P left parenthesis straight F with bar on top right parenthesis thin space straight P left parenthesis straight E with bar on top right parenthesis thin space straight P left parenthesis straight F right parenthesis space plus.......

equals space 8 over 9 cross times 1 over 6 plus 8 over 9 cross times 5 over 6 cross times 8 over 9 cross times 1 over 6 plus 8 over 9 cross times 5 over 6 cross times 8 over 9 cross times 5 over 6 cross times 8 over 9 cross times 1 over 6 plus..... equals space 8 over 9 cross times 1 over 6 open square brackets 1 plus space open parentheses 5 over 6 cross times 8 over 9 close parentheses space plus space open parentheses 5 over 6 cross times 8 over 9 close parentheses squared plus..... close square brackets equals space 8 over 9 cross times space 1 over 6 open square brackets fraction numerator 1 over denominator 1 minus begin display style 5 over 6 end style cross times begin display style 8 over 9 end style end fraction close square brackets space space space space space space space space space space space space space space space space space space space space space space space open square brackets because space space straight S space equals space fraction numerator straight a over denominator 1 minus straight r end fraction close square brackets equals space 8 over 9 cross times 1 over 6 open square brackets fraction numerator 6 space cross times space 9 over denominator 6 space cross times 9 space minus space 5 cross times 8 end fraction close square brackets space equals space 8 over 9 cross times 1 over 6 open square brackets fraction numerator 6 space cross times space 9 over denominator 14 end fraction close square brackets space equals space 8 over 9 cross times space 1 over 6 cross times fraction numerator 6 space cross times space 9 over denominator 14 end fraction equals space 4 over 7

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Probability

A and B throw two dice simultaneously turn by turn. A will win if he throws a total of 5, B will win if he throws a doublet. Find the probability that B will win the game, though A started the game.

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