Halving card piles game with a 1000-sided die

A stack of $n$ cards is prepared, where $n$ is the result of rolling a fair die numbered $1$ to $1000$. Alice and Bob then alternate turns, Alice moving first. On a turn, the player divides the current stack into two nonempty piles and throws away the smaller one (if the piles are equal, either may be thrown away), leaving the other as the new stack. A player who cannot divide the stack loses. If both play perfectly, what is the probability that Alice wins?

$991/1000$. From $n$ cards you can leave any stack size from $\lceil n/2\rceil$ to $n-1$. By induction the losing sizes are exactly $2^k-1$: from $2^k-1$ every reachable size lies strictly between $2^{k-1}-1$ and $2^k-1$ (all winning), while any other $n$ can reach the losing size just below it. There are $9$ such values up to $1000$ ($1,3,\dots,511$), so Alice wins in the other $991$ cases.