Fair die plus a value-weighted die: odd total

Emma throws two six-sided dice numbered $1$ to $6$. The first is fair. The second is biased so that the chance of each face is proportional to the number on it. What is the probability that the two numbers add to an odd total?

$1/2$. An odd sum needs exactly one die to be odd. Because the fair die is odd or even with probability $\tfrac12$ each, $P(\text{odd sum}) = \tfrac12 P(\text{biased odd}) + \tfrac12 P(\text{biased even}) = \tfrac12$, whatever the bias.