The secretary problem, random walk hitting times, and Wald's identity E[S_τ]=E[X]·E[τ]. Explore the 1/e rule for the secretary problem and optimal stopping regions for random walks.
Secretary problem: reject first r* ≈ N/e candidates (observe only), then take first who beats all seen — optimal success probability → 1/e ≈ 37%. Wald's identity: if τ is stopping time ind. of X_i, then E[Σ_{i=1}^τ X_i] = E[X]·E[τ]. Random walk hitting time: τ_b=inf{n: S_n=b}, E[τ_b]=b/μ (drift μ>0), Var[τ_b]=b·σ²/μ³.