The binomial and Poisson count how many. The geometric flips the question to how long: how many trials until the first success? For a sales floor dialing leads, that is the number of calls to the first win, and a single rate p governs the whole waiting game.
telemarketing_sales_outreach.csv records calls_to_first_success and a
lead_segment for 1,000 outreach runs. We model the calls-to-win as geometric and ask: what is the
success rate, which segment is easiest, and how should reps budget their calls?
Fit the Success Rate
The geometric mean is 1/p, so the per-call success rate is one over the average wait: p = 1/6.45 = 0.155. That single number reproduces the whole shape, and its large variance, (1−p)/p² = 35, reflects the long right tail of stubborn leads.
The shape is unmistakable: a win on the first call is the single most likely outcome, yet the slowly-decaying tail of hard leads pulls the average up to 6.5 calls. That tension, a likely quick win but a heavy tail, is the signature of the geometric.
Which Segment Is Easiest?
Each lead segment is its own geometric process. Comparing the per-call rate p = 1/mean across segments is a direct read on lead quality, and a lever for routing reps.
| Segment | mean calls to win | per-call rate p | records |
|---|---|---|---|
| SMB | 6.27 | 0.159 (easiest) | 494 |
| Mid-Market | 6.37 | 0.157 | 402 |
| Enterprise | 7.59 | 0.132 (hardest) | 104 |
SMB leads convert fastest, about 6.3 calls per win, while Enterprise takes nearly 7.6, a lower per-call rate that reflects longer, more complex sales. A manager can act on this immediately: weight call budgets and training by each segment's p, rather than treating every lead as equal.
Memoryless: the Sunk-Cost Trap
The geometric has a famous, counterintuitive property: it is memoryless. After any number of failures, the expected number of remaining calls to a win is still 1/p. Simulating two million runs confirms it.
Whether a rep is on call 1 or has just failed 5 times, the expected remaining wait is the same 6.5 calls (the simulation gives 6.45 then 6.46). A streak of rejections does not make the next yes more likely, the gambler's fallacy in a sales suit. There is a 57% chance of a win within 5 calls and 82% within 10, so budgets should come from p, not from a hot or cold streak.
Run the outreach analysis
The companion notebook fits the per-call rate from the mean wait, overlays the geometric PMF on the observed calls-to-win, validates the fit with a chi-square goodness-of-fit test, breaks the rate down by lead segment, and simulates two million runs to demonstrate the memoryless property and the win-within-k probabilities, every figure from code.
View opens the rendered notebook instantly (no setup). Open in Colab runs &
edits it live in your browser. To run locally, install numpy, pandas, scipy,
and matplotlib and launch jupyter notebook.
🎓 Key Takeaways
- ✓Calls to the first win are geometric: PMF (1−p)k−1p, mean 1/p, a long right tail.
- ✓Fit p from the mean: p = 1/6.45 = 0.155, with large variance (1−p)/p² = 35.
- ✓Each segment has its own p: SMB easiest (0.159), Enterprise hardest (0.132).
- ✓Memoryless: expected remaining calls stays 1/p ≈ 6.5 no matter how many failures came before.
- ✓Budget from p, not streaks: 57% of wins land within 5 calls, 82% within 10.
Quiz: Test Yourself
Eight quick questions on the geometric case study. Answer them, hit Check Answers, and keep refining until you score 100%. Your progress is saved, so you can hop back to the chapter and return anytime.