There are a number of chess programs that will calculate win % based on any board arrangement. Is there a specific opening (first move) for white or black that is best from a probabilistic standpoint?
I'm guessing there is for white, and the optimal black opening will depend on what move has been made prior. I'm trying to make no presumption on the skill level of players from which these probabilities have been calculated.
From a game-theory standpoint, Zermelo's theorem states that for each opening, one of the following must be true:
This theorem applies to any finite, two-player game with perfect information, where players alternate moves and there is no element of chance. Therefore, if Chess is ever weakly-solved, we will find the optimal first moves and they will depend on whether the solution is a White win, Black win, or a draw:
From a probabilistic standpoint, we can analyze the win/lose/draw statistics of games that started with each opening. However, the skill of the players is crucial. I am not a Chess expert, but I know that some openings are easier than others. What I mean is that the win/lose/draw statistics for a particular opening can vary significantly depending on whether the games are played by amateurs or masters.
This question is vague, so the answer itself is unclear. At the risk of giving you more than you wanted, here're some answers anyway.
If you take "best from a probabilistic standpoint" to mean that the game-theoretic result is the best one possible, and all winning positions are equivalent (and all drawn positions are equivalent), then the answer is "no". That's because chess is almost surely a draw, so unless your opening move is so bad you lose immediately afterwards,* your first move is just a choice between different options that are equally good. In that sense 1. e4, 1. d4, 1. c4, and 1. b4 are all equivalent.
If on the other hand you take "best from a probabilistic standpoint" as the opening evaluation of the strongest chess engines today, then the best moves are likely 1. e4 and 1. d4. These are the two moves that engines are most likely to choose, if left to their own devices, on move 1. The exact "best move" depends on several factors however, such as 1) hardware used 2) time you let the engine think for and 3) which version of the engine is used (different versions of the same engine prefer different moves).
On yet another hand, if you take the "best from a probabilistic standpoint" as the move that wins the most in chess databases, then the best move depends on the database you use, how much importance you ascribe to statistical significance, and whether you impose any cutoffs like "master games only". Doing neither of the latter two means 1. Na3 is best according to these two databases. You write in the OP that "I'm trying to make no presumption on the skill level of players" - but if you take this literally, then worstfish** is a legitimate player, and you can easily skew your results by getting worstfish to play against random_move.
So what does "best from a probabilistic standpoint" mean to you? The answer will differ for different definitions.
*There is some circumstantial evidence that 1. g4 is the only first move that is losing for White.
**This is Stockfish configured to play the worst move available.
The current consensus is that chess will not be solved, at least not any time soon. Because the game isn't be solved, that means there is no single optimal opening move for white, and even more so for black.
Chess is one of the games where you play your opponent as much as you play the game, different styles, strategies change how good each possible opening move is because of how players will go through the rest of the game. The Scholars Mate for example is a great opening move against very new players because it leads to the fastest possible win, but against someone who is experienced it's overextending valuable pieces without significant defenses.
White is making their opening move on a single possible board state, black on the other hand has 20 possible board states (8 white pawns can each move up one or two makes 16, 2 knights can either move left or right for the other four), and each of those board states would have moves that are better or are worse based on what white has access to and exposed with their first move.
