  In this article, we study the behavior of a broad family of real sequences
derived from randomized one-pile subtraction games. For any subtraction set
$S$, we allow any valid number of chips $s\in S$ to be removed at equal
probability at any given position and we study the sequences
$(a_n^S)_{n\in\mathbb{N}}$ representing the probability of winning the game
from a position with $n$ chips. We characterize these sequences in terms of
linear recurrence relations and examine their behavior as $n\rightarrow\infty$
for all finite $S$. We fully solve the cases for subtraction sets of fewer than
3 elements and partially complete the general case for arbitrary $S$.
