# Odds of 100% first turn accusation in Clue?

Last night my family played clue for the first time in years. It was a 4-person game, though i don't think that matters for this question. I went first and rolled high enough to get to a room for which i didn't have the card. I picked a person and weapon i didn't have as well.

No one had any of the three, so on my first move I'd guessed the right answer.

Obviously the odds of guessing if you are unconstrained relate to 1 in 6, 1 in 6, 1 in 9. Probably 1-(5/6 × 5/6 × 8/9). But then there are the odds of being placed near the correct room and rolling high enough to get there.

Anyone work that part out already?

Edit: the complexity comes entirely from the number of players, the locations of the pieces they get, and the rolls (which in some cases can get you to two rooms from the starting position), and whether they have that room already. Then you make a suggestion, which has to miss everyone.

• The answer here depends on what's in your hand - for example, if the only room you can reach on your first turn is one in your hand, then your chance of making a successful accusation is zero. – Philip Kendall May 21 '17 at 15:36
• Of course that's why I'm saying, you could consider all card distributions and layouts and die rolls and compute it. I am guessing someone has beaten me to it :). But it's a cute problem, I think. – Fred Douglis May 21 '17 at 15:38
• Die rolls are actually mostly irrelevant - Accusations do not require you to be in the room you name. The only affect a die roll can have is whether you get the opportunity to have a suggestion BEFORE you make your accusation (which can help narrow it down), and in that case, having the relevant room card in your hand is actually better since it manipulates what information you get. – KitsuneZeta May 21 '17 at 16:00
• The die roll was to get to the room for the suggestion. – Fred Douglis May 21 '17 at 16:37
• making a suggestion is not a prerequisite for making an accusation, and in some versions of the rules, having a suggestion proved false blocks you from making an accusation (Hasbro's official ruling on this is the opposite - you may always make an accusation after a suggestion, even if the suggestion was proven false). Also, I was wrong on having the room card in your hand improving odds. – KitsuneZeta May 21 '17 at 17:02

# 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
• Thanks for the analysis... interesting! I'll give people a chance to comment/correct before marking it as correct, but it seems like what I was getting at. I'm not entirely sure... since you're assuming getting dealt a helpful hand, whereas I'm talking cumulative. – Fred Douglis May 21 '17 at 22:44