Observation of the Standard Model Higgs boson produced in association with a pair of top quarks at $\sqrt{s} = 13 \, \text{TeV}$ with the ATLAS experiment at the LHC with emphasis on the decay of the Higgs boson into a $b\bar{b}$-pair in the single-lepton channel

The top quark is the heaviest elementary particle in the Standard Model and has an expected Yukawa coupling to the Higgs boson of order unity. The value of this coupling is a key ingredient to unravel the nature of the observed Higgs boson. The most favourable production mode that has a direct sensitivity to this coupling is the production of a Higgs boson in association with a top-quark pair, $t\bar{t}H$. This process was observed based on the analysis of proton-proton collision data at $\sqrt{s} = 13 \, \text{TeV}$ recorded with the ATLAS experiment at the LHC. Using data corresponding to integrated luminosities of up to $79.8 \, \text{fb}^{-1}$, and considering the Higgs boson decays into $b\bar{b}$, $WW^{*}$, $\tau^{+} \tau^{-}$, $\gamma \gamma$, and $ZZ^{*}$ yields a signal strength of \begin{equation*} \mu = 1.32 \pm 0.18 (\text{stat.})^{+0.21}_{-0.19}(\text{syst.}) = 1.32^{+0.28}_{-0.26}, \end{equation*} corresponding to an observed (expected) signal significance of 5.8 (4.9) standard deviations. The analysis targeting the Higgs boson decay channel with the highest branching ratio, $t\bar{t}H (H \rightarrow b\bar{b})$, uses data corresponding to an integrated luminosity of $36.1 \, \text{fb}^{-1}$ and will be presented in detail. A focus is placed on the single-lepton channel. The dominant background for this channel is $t\bar{t}b\bar{b}$. One of the small backgrounds originates from non-prompt leptons and fake leptons, which originate from jets misidentified as a reconstructed lepton. This background requires a special treatment in signal regions with many jets and $b$-jets. Despite its small contribution, an estimate of non-prompt leptons and fake leptons is important for a successful measurement in the $t\bar{t}H$ analysis as well as other analyses with leptonic final states. In this thesis, a fully data-driven technique - the matrix method - is presented. For the first time, efficiencies for the 2017 dataset are shown. In addition, a tag rate function could be employed to increase the performance of the matrix method for a fixed $b$-tagging working point. Finally, the performance of neural networks using low-level input variables is examined to discriminate the $t\bar{t}H$ signal from backgrounds.