When lead nitrate (Pb(NO$_3$)$_2$) and calcium iodide (CaI$_2$) are combined, a chemical reaction takes place, resulting in the formation of lead iodide (PbI$_2$) and calcium nitrate (Ca(NO$_3$)$_2$). The balanced chemical equation for this reaction is:

Pb(NO$_3$)$_2$ + CaI$_2$ → PbI$_2$ + Ca(NO$_3$)$_2$

In this reaction, the lead ions (Pb$^{2+}$) from lead nitrate combine with the iodide ions (I$^-$) from calcium iodide to form lead iodide, which is a yellow precipitate. Simultaneously, the calcium ions (Ca$^{2+}$) from calcium iodide combine with the nitrate ions (NO$_3^-$) from lead nitrate to form calcium nitrate, which remains dissolved in the solution.