$a$ is chosen from the set $\{1,2,3,4\}$, and $b$ is chosen from the set $\{5,6,7,8,9\}$. What is the probability that the product $ab$ is a prime number?
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 $\frac14$. Furthermore, $b$ must be prime, so $b$ must equal 5 or 7, and this occurs with probability $\frac25$. Therefore, the probability that $ab$ is prime is $\frac14\cdot \frac25 = \boxed{\frac1{10}}$.