jmc

It is easier to count the number of integers from 1 to 150 that are perfect powers. We see there are 12 perfect squares from 1 to 150, namely $1^2,2^2,\dots ,12^2$, and there are 5 perfect cubes, namely $1^3,\dots ,5^3$. Notice all the perfect fourth powers are also perfect squares. Similarly, all the perfect sixth powers are also perfect squares. The only perfect powers not yet counted are $2^5=32$ and $2^7=128$. Then notice there are two repetitions, $1^6=1$ and $2^6=64$ which we counted both as perfect squares and perfect cubes. So there is a total $12+5+1+1-2=17$ of integers from 1 to 150 that are perfect powers. Thus, $150-17=133$ integers are not perfect powers. The probability that we select such a number is $\boxed{\frac{133}{150}}$.