jmc

The product $ab$ is prime only when either $a$ or $b$ is one. Since $b$ cannot equal 1, $a$ must equal 1, and this occurs with probability $\frac{1}{4}$. Furthermore, $b$ must be prime, so $b$ must equal 5 or 7, and this occurs with probability $\frac{2}{5}$. Therefore, the probability that $ab$ is prime is $\frac{1}{4}\cdot \frac{2}{5}=\boxed{\frac{1}{10}}$.