Precision tests of the Standard Model using key observables of $CP$ violation and rare decays

In this thesis advanced statistical methods are used for precision studies in the flavour sector of the Standard Model of particle physics. The necessary tools are developed and applied to two key measurements of the LHCb experiment: the determination of the CKM angle $\gamma$ and the search for rare $B^{0} _{s} \to \mu^+ \mu^-$ and $B^0 \to \mu^+ \mu^-$ decays. The CKM angle $\gamma$ is, for the first time, measured from $B^0 _s \to D^\mp _s K^\pm$ decays using a dataset corresponding to $1\,\mathrm{fb}^{-1}$ of $pp$ interactions at a centre-of-mass energy of $\sqrt{s} = 7\mathrm{TeV}$. The result of $\gamma = (115^{+27}_{-43})^{\circ}$ is then combined with a set of $\gamma$ measurements in ${B} \to Dh$ decays resulting in a precision on $\gamma$ of $<8^{\circ}$. This result improves the legacy results from the $B$-factories by more than a factor of two. The rare decays $B^{0} _{s} \to \mu^+ \mu^-$ and $B^0 \to \mu^+ \mu^-$ are analysed on a dataset corresponding to $3\,\mathrm{fb}^{-1}$ of $pp$ interactions at a centre-of-mass energy of $\sqrt{s}=7\text{ and }8\,\mathrm{TeV}$. The branching fraction of $B^{0} _{s} \to \mu^+ \mu^-$ decays is measured to be consistent with the Standard Model prediction, while no significant excess of $B^0 \to \mu^+ \mu^-$ events is found. Then, the LHCb and CMS datasets from the first run of the LHC are combined to perform a joint search for $B$ meson decays into two muons. The decay $B^{0} _{s} \to \mu^+ \mu^-$ is observed for the first time with a statistical significance of $6\sigma$ and the first evidence for $B^0 \to \mu^+ \mu^-$ decays is found with a significances of $3\sigma$.