= Solution
Let $C_{12}=\mathbb E[S_1S_2]\geq0$, using the supplied correlation property of the <Curie–Weiss model>. Since $\mathbf1_{\{S_i=1\}}=(1+S_i)/2$ and each spin has zero <expected value>,
$$
\mathbb P(S_1=1,S_2=1)
=\frac14\mathbb E[(1+S_1)(1+S_2)]=\frac{1+C_{12}}4.
$$
Divide by $\mathbb P(S_1=1)=1/2$ to obtain
$$
\boxed{\mathbb P(S_2=1\mid S_1=1)=\frac{1+C_{12}}2
\geq\frac12=\mathbb P(S_2=1).}
$$
The expression assumes $n\geq2$, as required for $S_2$ to exist. It makes the connection between the spin correlation and the requested <conditional probability> explicit.
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