Quantify base-rate dilution in CTR

The overall click-through rate is a volume-weighted average over the different notification categories. Let the baseline non-event notifications have a click-through rate of $b = 2.0\%$ and currently make up $N$ messages. A proposed event notification has a click-through rate of $e = 20.0\%$ but starts with only $k$ messages. 1) Give the expression for the overall CTR: $C_{overall} = \frac{bN + ek}{N+k}$. 2) Using $N = 9,990$ and $k = 10$, calculate $C_{overall}$ and its relative lift over the baseline, then explain why a 10× click-through rate on such a small volume creates a negligible shift in the overall metric. 3) Derive the smallest share $s = \frac{k}{N+k}$ needed to produce at least a $10\%$ relative lift in overall CTR, meaning $C_{overall} \ge 1.10b$, written in terms of $b$ and $e$; then substitute $b = 2.0\%$ and $e = 20.0\%$ to find $s$. 4) Under a send-budget limit, determine the smallest integer $k$ that satisfies the target when $N$ is fixed at $1,000,000$.

Overview: The question tests how well a candidate grasps weighted averages, base-rate or dilution effects, and the algebra of proportions when combining different click-through rates across subpopulations.