Question

Sofia and Tess will each randomly choose one of the 10 integers from 1 to 10. What is the probability that neither integer chosen will be the square of the other?

Answer 1

There are $10^2=100$ possible pairs they can choose. Let $S$ and $T$ denote the integers that Sofia and Tess, respectively, choose, so that $(S,T)$ denotes the integer pair.

The event that either $S=T^2$ or $S^2=T$ has 5 possible ways of occurring:

$\{(1,1),(2,4),(4,2),(3,9),(9,3)\}$

So the probability of this happening is $\dfrac5{100}=\dfrac1{20}=0.05=5\%$. You're interested in the complement, which means the answer is 95%.