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Halve or diminish, and race to unity! v2

This is follow-up question to Halve or diminish, and race to unity!.

Alice and Bob are playing a game. In the beginning, they randomly choose an positive odd integer between $2$ to $n$ where $n$ is a finite number. In a move, a player, can:

  • either decrease the number on the board by $1$ (i.e., replace $n$ by $n-1$), or
  • halve the number, and round it up to the next integer if a fraction is obtained (i.e., replace $n$ by $\left\lceil \frac{n}{2}\right\rceil$).

As always, Alice moves first, and then they make moves in turns. The first person to get to $1$ wins.

Assuming both players play perfectly, who has the better chance to win and what is that chance?