Suppose one generates a random polynomial of degree $d$ with integer coefficients uniformly distributed within $[-c_\max,c_\max]$. For example, for $d=8$, $|c_\max|=100$, here is one random polynomial: $$ -46 x^8-19 x^7+14 x^6-75 x^5+94 x^4+18 x^3-48 x^2+29 x-61=0 \;. $$
Q. What is the probability that such a random $(d,c_\max)$-polynomial has at least one integer solution?
Here is a bit of data, based on $10^5$ polynomials for each $d=1,\ldots,10$ with $|c_\max|=100$:
![DiophantinePoly][1]
I.e., For $d=1$, about $5.5$% have integer solutions, while for $d=10$, about $0.8$% have integer solutions. The above displayed degree-$8$ example has the solution $x=-1$., and in addition these $7$ non-integer roots: $$ -1.54767,\\ -0.0337862 \pm 0.794431 i,\\ 0.196632 \pm 1.19591 i,\\ 0.904467 \pm 0.323333 i \;. $$
If the constant coefficient is $0$, then of course $x=0$ is a solution. So $\frac{1}{2 c_\max+1}$ is a lower bound, in the above chart, $\frac{1}{201} \approx 0.5$%.