The best odds possible for a turn 1 win without a blind accusation in a four-player game are 7/648 as Mrs. Peacock holding 5 cards.
There are two probabilities at play here. First, the probability of reaching a room on turn one, which is not the same for all players, and then second, the probability of the resulting suggestion being correct on all counts (that is, no cards are shown to the player and [barring one exception] no cards in hand were used in the suggestion), which is dependent exclusively on the cards that player holds in their hands. We'll first look at the suggestion part of this, since that's where the bulk of the math lies.
The best possible opening hand to get this result is either all suspect cards or all weapon cards.
The odds for the first suggestion of the game being correct mostly depend on the cards the player making the suggestion has in their hand. There are 6 Suspects, 6 Weapons, and 9 Rooms. The effective odds for a player making a suggestion (presumably in a room they don't have the card for) can be determined by 1 / ((6 - held suspect cards) * (6 - held weapon cards) * (9 - held rooms)). The best case for this ends up being a situation where a player holds a hand consisting of either all suspect cards OR all weapon cards. In a 3-player game, in which each player is given 6 cards, the 6th card should not be a room (i.e. the best hands are 5 suspects and 1 weapon or 5 weapons and 1 suspect). Following this, the odds of a correct suggestion (or a blind accusation, for that matter) given the best starting hand and no other information are as follows:
- 3 Players: 1/45 (6 cards)
- 4 Players: 1/54 (assuming 5 cards)
- 5 Players: 1/108 (assuming 4 cards)
- 6 Players: 1/162 (3 cards)
In a 4- or 3-player game, it is possible that a player may know either the Suspect or the Weapon just from their hand. In that case, it won't matter what they choose for that part of the suggestion, as your accusation does NOT have to be the same as your immediately-prior suggestion. In the case of knowing the suspect in particular, it is beneficial to name themselves in the suggestion so as not to give any other player a free suggestion on their turn by moving them to the room.
While the remainder of this answer simply asserts we did get this best hand for simplicity of math, the probability of actually being dealt said best hand for a given hand size, for all cases other than a 3-player game, are as follows:
- 5-card hand (4 players): 240/1,028,160 (or 1/4284)
- 4-card hand (5 or 4 players): 240/73,440 (or 1/306)
- 3-card hand (6 or 5 players): 120/4896 (or 10/408)
Having the best hand improves your odds of making the suggestion, but the only hard requirement would be not having the room cards for the rooms your pawn can reach on a roll of 12.
Not all players have the same odds of making a turn 1 suggestion
I will preface this section by noting that there are a few variants of the board that still have the rooms in the same spots, but do NOT have doors in the same spots. I'm using the most common board here for reference, which has all the players capable of entering a room with a move roll of 8. The roll required to reach the room closest to each character's starting position (and the room itself) are as follows:
- Ms. White: 8 (Ballroom)
- Mr. Green: 8 (Ballroom)
- Mrs. Peacock: 7 (Conservatory)
- Prof. Plum: 8 (Study)
- Ms. Scarlet: 8 (Lounge)
- Col. Mustard: 8 (Lounge OR Dining Room)
Col. Mustard stands out as having two different rooms he can reach with a roll of at least 8, so even if Col. Mustard has one (and only one) of those two rooms in his hand, he can just go to the other one. This isn't particularly relevant to our calculations, as Mrs. Peacock actually has the best odds of reaching a room on her initial move roll, with odds of 21/36 (as opposed to the odds of 15/36 that everyone else has). Furthermore, the best possible hand for this doesn't include any room cards.
With that, asserting that the best hand possible was dealt for this situation, for Mrs. Peacock, the odds for reaching a room on turn 1, making a suggestion, and the suggestion being correct are as follows:
- for three players, 7/540
- for four players, 7/648
- for five players, 7/1296
- for six players, 7/1944