Dice product with an optional single re-roll

(a) Two fair dice are rolled. What is the expected product of the two numbers?

(b) Now, after seeing the first roll, you may choose one of the two dice and roll it again (or keep both). If you want the product to be as large as possible, what is its expected value under the best strategy?

(a) $12.25$; (b) $617/36\approx17.14$. (a) Independence gives $3.5^2$. (b) Re-rolling the smaller die turns its value into an expected 3.5, so the outcome $(a,b)$ is worth $\max(ab,\,3.5\max(a,b))$; averaging this over the 36 rolls gives $\tfrac{617}{36}$.