= Solution
The <diffusion generator> is $\mathcal Lf=\tfrac12\sigma^2f''+bf'$. On a compact subinterval of $I$, the coefficient $s'\sigma$ is bounded, and the <Itô formula> up to its exit time $\rho$ gives
$$
\boxed{s(X_{t\wedge\rho})=s(X_0)+\int_0^{t\wedge\rho}s'(X_u)\sigma(X_u)dB_u,}
$$
because $\mathcal Ls=0$. Increasing compact subintervals proves the local-martingale property up to $T$. The notation $s(X_{t\wedge T})$ at an exit endpoint is understood with its continuous endpoint extension when needed; stopping on $[a,b]\subset I$ below is always well-defined. This is the <martingale> property of a <scale function of a one-dimensional diffusion>.
For the hitting ratio there are two necessary qualifications: $s(b)\ne s(a)$ and $\tau=T_a\wedge T_b<\infty$ almost surely. A usual scale function is strictly monotone, which ensures the first condition, but the printed hypotheses alone ensure neither. Under these conditions, $s(X_{t\wedge\tau})$ is a bounded <local martingale> and hence a <uniformly integrable> <martingale>. The <optional stopping theorem> gives
$$
s(x)=\mathbb Es(X_\tau)=s(a)\mathbb P(T_a<T_b)+s(b)\mathbb P(T_b<T_a).
$$
Continuity prevents simultaneous hitting of distinct endpoints, and almost sure exit makes these two probabilities sum to one. Solving yields the <boundary hitting probability from a diffusion scale function>
$$
\boxed{\mathbb P(T_b<T_a)=\frac{s(x)-s(a)}{s(b)-s(a)}\quad\text{when exit is a.s. finite and }s(b)\ne s(a).}
$$
The <hypotheses for a diffusion scale hitting formula> cannot be omitted here. Take $\sigma\equiv0$, drift $b\equiv0$, $s(y)=y$ and $X_t\equiv x\in(a,b)$. All printed coefficient and differential conditions hold, but neither endpoint is ever reached, so the left side is zero whereas the displayed ratio is strictly positive. A constant $s$ is another allowed solution of the printed differential equation and makes its denominator zero. In general the bounded stopped <martingale> has a limit $Y_\infty$, and optional stopping instead gives
$$
s(x)=s(a)\mathbb P(T_a<T_b)+s(b)\mathbb P(T_b<T_a)+\mathbb E[Y_\infty\mathbf1_{\{\tau=\infty\}}].
$$
The following parts have nondegenerate diffusion and strictly monotone powers, and their finite-exit properties are justified explicitly.
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