jmc

When two dice are rolled, there are 36 total outcomes. Let's compute the probability that Allen wins. Allen wins if the product of the two numbers is even and not a multiple of 3. In other words, Allen wins if the product is 2 $(1\cdot 2,2\cdot 1)$, 4 $(1\cdot 4,4\cdot 1,2\cdot 2)$, 8 $(2\cdot 4,4\cdot 2)$, 10 $(2\cdot 5,5\cdot 2)$, 16 $(4\cdot 4)$, or 20 $(4\cdot 5,5\cdot 4)$. Therefore, the probability that Allen wins is $\frac{2+3+2+2+1+2}{36}=12/36=1/3$. Then, the probability that Jean wins is $1-1/3=\boxed{\frac{2}{3}}$.