Two standard six-faced dice are rolled. Jean wins if the product of the two numbers rolled is odd or a multiple of three, otherwise Allen wins. What is the probability that Jean wins? Express your answer as a common fraction.
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 $(1\cdot2, 2\cdot1)$, 4 $(1\cdot4, 4\cdot1, 2\cdot2)$, 8 $(2\cdot4, 4\cdot2)$, 10 $(2\cdot5, 5\cdot2)$, 16 $(4\cdot4)$, or 20 $(4\cdot5, 5\cdot4)$. 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 $1-1/3=\boxed{\frac{2}{3}}$.