Quantify base-rate dilution in CTR

Overall CTR is a volume-weighted average across notification types. Suppose baseline non-event notifications have CTR b = 2.0% and currently account for N messages. A proposed event notification has CTR e = 20.0% but initially accounts for only k messages. 1) Write the formula for overall CTR: C_overall = (b·N + e·k)/(N + k). 2) With N = 9,990 and k = 10, compute C_overall and the relative lift vs baseline; explain why a 10× CTR on a tiny volume yields a negligible overall change. 3) Derive the minimum share s = k/(N + k) required to achieve at least a 10% relative increase in overall CTR (i.e., C_overall ≥ 1.10·b), expressed in terms of b and e; then plug in b = 2.0%, e = 20.0% to solve for s. 4) Given a send-budget constraint, solve for the smallest integer k that meets the target when N is fixed at 1,000,000.