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Two people, $P$ and $Q$, decide to independently roll two identical dice, each with $6$ faces, numbered $1$ to $6$. The person with the lower number wins. In case of a tie, they roll the dice repeatedly until there is no tie. Define a trial as a throw of the dice by $P$ and $Q.$ Assume that all $6$ numbers on each dice are equi-probable and that all trials are independent. The probability (rounded to $3$ decimal places) that one of them wins on the third trial is ____
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