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Problem 636

AMC 10/12, early questions

Algebra Difficulty 3.8 Find the answer China Mathematical Competition · China

Two persons roll two dice in turn. Whoever gets the sum number greater than $6$ first will win the game. The probability for the person rolling first to win is ________.

Your answer

A number or a short expression. Fractions can be typed as 3/2, and spacing doesn't matter.

Official solution

The probability for rolling two dice to get the sum number greater than $6$ is $\frac{21}{36} = \frac{7}{12}$ . Therefore, the required probability is
$\frac{7}{12} + \left(\frac{5}{12}\right)^2 \frac{7}{12} + \left(\frac{5}{12}\right)^4 \frac{7}{12} + \cdots = \frac{7}{12} \times \frac{1}{1 - \frac{25}{144}} = \frac{12}{17}.$

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