First considering without modifiers
When you have advantage the probability of getting a result of X can be governed by the equation
$$
P(x)= \frac {2x-1} {400}
$$
When you have disadvantage the probability of getting a result of X can be governed by the equation
$$
P(z)= \frac {41-2z} {400}
$$
The overall Probability of winning when you roll x, with advantage vs disadvantage, that you roll X is given by:
$$ P(\text{Winning by rolling } x) = P(x) \times P(z < x) $$
$$ P(z < x) = f(x) $$
With a little looking we see that each probability for \$P(z)\$ has the common thread of a reduction by 0.005
Such that \$P(z=2) = P(z=1)-0.005\$ and \$P(z=3) = P(z=2)-0.005\$
For any particular \$x\$ we want to sum P(z) over z=1 to z=x-1
From this we get that:
$$
P(z < x) = 0.0975(x-1) - 0.005 \dfrac {(x-1)(x-2)} {2} \qquad \text{where 1 =< x =< 20}
$$
(The \$\frac{(x-1)(x-2)}{2}\$ term simply denotes multiplying by a series of triangle numbers, starting at \$1\$ when \$x=3\$)
Hence we get:
$$
P(\text{Winning by rolling } x)= \dfrac{\left(2x-1\right)\left(0.0975(x-1)-\left(0.005 \dfrac{(x-1)(x-2)}{2}\right)\right)} {400}
$$
or alternately:
$$
P(x)=-0.0000125(x-40)(x-0.5)x
$$
Now we can sum over all values of \$x\$ to get \$P(\text{Winning})\$ for all \$x\$
$$
P(\text{Winning}) = 0.8158125
$$
Now we consider modifiers
By including modifiers we're no longer summing from \$z=1\$ to \$z=x-1\$.
Instead we have to sum from \$z=2-m\$ to \$z=x+m-1\$. Here we hit trouble. If \$2-m < 1\$ then those terms won't be valid. If \$x+m-1 > 20\$ then, again those terms won't be valid.
We, however, want that if \$z < 1\$ that the formula \$f(z) = 0\$ and that if \$z > 20\$ that \$f(z) = 1\$.
\$2-m < 1\$ occurs when \$m > 1\$ and that \$x+m-1 > 20\$ when \$m > 2\$
Without a good way to enforce these restrictions in-formula we're forced to give the following:
\begin{align}
f(z) &= \dfrac{41-2z}{400} & \text{for 1 =< z =< 20} \\
f(z) &= 0 & \text{for z < 1} \\
f(z) &= 1 & \text{for z < 20}
\end{align}
Hence we have:
$$
P(\text{Winning by rolling x})= \sum\limits_{a=2-m}^{a=x+m-1} f(a) \times \frac{2x-1}{400}
$$
Previously from here we could produce a formula for the P(Winning). Here, however, we cannot algebraically sum for all values of x in a way that can be simplified. Instead we get the following:
$$
P(\text{Winning})= \sum\limits_{x=1}^{x=20} (\sum\limits_{a=2-m}^{a=x+m-1} f(a) \times \frac{2x-1}{400})
$$
However, since our sums behave nicely for \$m < 0\$ we can create the following:
$$
P(\text{Winning}) = (783180+34399 m-2278 m^2-82 m^3+m^4)/960000 \qquad \text{ for m < 0}
$$
Graph of Probability of Winning as function of m:
