jmc

It is easier to count the number of integers from 1 to 150 that are perfect squares or perfect cubes. 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 $1^3,\dots ,5^3$. Then notice there are two repetitions, $1^6=1^2=1^3=1$ and $2^6=8^2=4^3=64$. So there is a total of $12+5-2=15$ integers from 1 to 150 that are perfect squares or perfect cubes. Thus, we get $150-15=135$ integers from 1 to 150 that are neither perfect squares nor perfect cubes. So the probability that we select such a number is $\frac{135}{150}=\boxed{\frac{9}{10}}$.