Orange ball after moving 300 balls between urns

Urn $X$ holds $400$ orange and $100$ yellow balls, and urn $Y$ holds $100$ orange and $400$ yellow balls. A random set of $300$ balls is taken from urn $X$ and added to urn $Y$. A single ball is then chosen uniformly at random from the $800$ balls now in urn $Y$. What is the probability it is orange?

$\tfrac{17}{40}$. The transferred balls are on average $80\%$ orange, so urn $Y$ expects $240+100=340$ orange balls out of $800$. By the law of total probability the chance of orange is $\tfrac38\cdot\tfrac45+\tfrac58\cdot\tfrac15=\tfrac{17}{40}$.