jmc

We will subtract the probability that the product is odd from 1 to get the probability that the product is even. In order for the product to be odd, we must have both numbers be odd. There are $2\cdot 2=4$ possibilities for this (a 3 or 5 is spun on the left spinner and a 5 or 7 on the right) out of a total of $3\cdot 4=12$ possibilities, so the probability that the product is odd is $4/12=1/3$. The probability that the product is even is $1-1/3=\boxed{\frac{2}{3}}$.