Theorem. Zero-hit likelihood under iid evaluation [ftip-00A1]
✍️source

Let \(m\in \mathbb N_{\geq 1}\). Assume the hit indicators from Definition [ftip-009Y] are independent Bernoulli variables with a common success probability \(p\). Then \(K_m\) has the binomial law and

\[ \Pr _p(K_m=0)=(1-p)^m. \]

Proof.

The event \(K_m=0\) is \(\bigcap _{j=1}^m\{I_j^\star =0\}\). Independence and \(\Pr _p(I_j^\star =0)=1-p\) give the displayed product.

This finite statement follows from the displayed hypotheses. Applying it to a source table requires the additional iid and stationary-success assumptions stated above.

Theorem. Zero-hit likelihood under iid evaluation [ftip-00A1]
✍️source

Proof.

Finite initial-coverage measurement [ftip-009X]

Definition. Initial evaluation hit count [ftip-009Y]

Definition. Empirical initial success rate [ftip-009Z]

Remark. Zero observed hits are not zero probability [ftip-00A0]

Theorem. Zero-hit likelihood under iid evaluation [ftip-00A1]
✍️source

Proof.

Corollary. Exact one-sided bound after zero hits [ftip-00A2]

Proof.

Example. The 128-sample zero-hit bound [ftip-00A3]

Definition. First positive sparse-reward time [ftip-00A4]

Corollary. Sparse-reward discovery within a rollout budget [ftip-00A5]

Proof.

Example. The AIME 128-sample measurement record [ftip-00A6]

Remark. An observed plateau is not a universal probability threshold [ftip-00A7]

Corollary. Exact one-sided bound after zero hits [ftip-00A2]

Proof.

Definition. Initial evaluation hit count [ftip-009Y]