Let $n$ smaller cubes of edge length $1 \mathrm{cm}$ are obtained after cutting bigger cube of edge length $5 \mathrm{cm}$.

Volume of bigger cube$=n\times$Volume of smaller cube

$\Rightarrow 5^3=n\times 1^3$

$\Rightarrow n=125$

When the painted cube is cute into $125$ small equal cubes, we have

 $3$ faces painted cube on every corner $\Rightarrow$Number of cubes with $3$ face painted$=1\times 8=8$

Three $2-$faces painted cube on every edge

$\Rightarrow$Number of cubes with 2 face painted$=3\times 12=36$

Nine single face painted cube on side $\Rightarrow$Number of cubes with $1$ face painted$=6\times 9=54$

Therefore, total number of cubes with any face painted$=8+36+54=98$

Let 

$A\to$Event that the the cube is painted

$B\to$Event that teh cube has three face painted

$P\left(A\cap B\right)=\frac{8}{125}$

$P\left(B\right)=\frac{98}{125}$

So, required probability is

$P\left(\frac{A}{B}\right)=\frac{P\left(A\cap B\right)}{P\left(B\right)}$

$\Rightarrow P\left(\frac{A}{B}\right)=\frac{{\displaystyle \frac{8}{125}}}{{\displaystyle \frac{98}{125}}}$

$\Rightarrow P\left(\frac{A}{B}\right)=\frac{8}{98}=\frac{4}{49}$