Detect-and-retry arithmetic

ProofIntermediate⏱ 30 minAgentsretryrecoverydetection

Problem

Prove $p_{\text{eff}} = 1-(1-p)(1-dp)^m$ for detection probability $d$ and $m$ retries.
For $p = 0.85$ , tabulate $p_{\text{eff}}$ and 30-step trajectory success over $d\in\{0,0.5,0.9,0.99\}$ with $m=3$ .
Show that as $m\to\infty$ , $p_{\text{eff}}\to1$ for any $d>0$ , and explain why this limit is misleading in practice.

Write a complete argument. Every inequality you use should be one you could state precisely.

Opening this before you have a real attempt on paper is the fastest way to feel like you understand something you cannot reproduce.

Problem	Field	Type	Difficulty	Time
Trajectory success from per-step reliability	Agents	Numerical	Foundation	15 min
The reliability wall	Agents	Numerical	Beginner	15 min
Majority voting via Hoeffding	Agents	Proof	Intermediate	30 min
What correlation does to voting	Agents	Proof	Intermediate	30 min
Simulate an agent pipeline end to end run	Agents	Coding	Advanced	1 day