Let $C$ be a projective smooth curve of genus $g$. Let $\Omega_C$ be the sheaf of differentials of $C$ and consider the exact sequence of vector bundles:
$$ 0\rightarrow \mathcal{O}_C\rightarrow E \rightarrow \Omega_C \rightarrow 0.$$
If $P(E)$ is the projective bundle over $C$ associated with $E$, the intersection number
$$(\mathcal{O}_{P(E)}(1) \cdot \mathcal{O}_{P(E)}(1))_{P(E)}$$
is well-known to be $2g-2$.
I have tried to prove this using intersection theory of Chow rings, but I obtain the number $2-2g$. Could anyone help me understanding where I make a mistake?
1) From the above exact sequence, we get that the Chern classes of $ E$ and $\Omega_C $ coincide.
2) $(\mathcal{O}_{P(E)}(1) \cdot \mathcal{O}_{P(E)}(1))_{P(E)}=\int_{P(E)}c_1(\mathcal{O}_{P(E)}(1))^2=\int_{C}s_1(E),$
where $c_1$ denotes the first Chern class and $s_1$ denotes the first Segre class.
3) $\int_{C}s_1(E)=-\int_{C}c_1(E)=-\int_{C}c_1(\Omega_C)=-deg(\Omega_C)=-(2g-2)$