Given positive integer $N$, we choose $m_1, m_2, n_1, n_2$ independently and with equal probabilities from $\{0,1,\ldots,N\}$, and let $f_1 = x^{m_1} + (1+x)^{n_1}$ and $f_2 = x^{m_2} + (1+x)^{n_2}$ as polynomials in the indeterminate $x$ over the field ${\mathbb F}_2$ of two elements. Let $P(N)$ be the probability that $f_1$ and $f_2$ are coprime. What can be said about $P(N)$, in particular its asymptotics as $N \to \infty$?

By explicit enumeration in Maple, the first few values are $$\eqalign{P \left( 1 \right) &={\frac {9}{16}},P \left( 2 \right) ={\frac {56}{ 81}},P \left( 3 \right) ={\frac {45}{64}},P \left( 4 \right) ={\frac { 489}{625}},P \left( 5 \right) ={\frac {1019}{1296}},\cr P \left( 6 \right) &={\frac {1895}{2401}},P \left( 7 \right) ={\frac {3299}{4096} },P \left( 8 \right) ={\frac {5308}{6561}},P \left( 9 \right) ={\frac {2023}{2500}},P(10) = \frac{11954}{14641}\cr}$$ Random sampling seems to indicate $P(100) \approx 0.83$. The sequence $(N+1)^4 P(N)$ does not appear to be in the OEIS.

EDIT: That sequence is now in the OEIS as A245488. $P(100) = 86648767/101^4 \approx 0.8326776196$.