Let $X$ and $Y$ be two independent random variables representing the results of two different dice. One die is fair, and the other is biased such that $P(6) = 0.5$ and $P(1)=P(2)=P(3)=P(4)=P(5) = 0.1$. Given that the sum of the dice is 7, what is the probability that the biased die showed a 6?