The amount of ways to assign 20 balls to 14 urns where each ball has at most 5 balls is the coefficient of x^20 in (1 + x + x^2 + x^3 + x^4 + x^5)^14, i.e.

310829610.

The amount of ways to assign 20 balls to 14 urns where each ball has at most 2 balls is the coefficient of x^20 in (1 + x + x^2)^14, i.e.

93093.

So the total probability is \(\displaystyle \frac{310829610-93093}{310829610} \approx 0.9997\)