Red cherry chance after moving half a bowl

Bowl $A$ contains $100$ red and $300$ purple cherries, while bowl $B$ contains $300$ red and $100$ purple cherries. Half of the cherries in bowl $A$, chosen at random, are moved into bowl $B$. A single cherry is then drawn at random from bowl $B$. What is the probability it is red?

$7/12$. After the move bowl $B$ holds $600$ cherries: $200$ came from $A$ (each red with probability $\tfrac14$) and $400$ were already there (red with probability $\tfrac34$). By total probability, $\tfrac13\cdot\tfrac14+\tfrac23\cdot\tfrac34=\tfrac{7}{12}$.