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When I was very young, my brother and I got the board game of Cluedo for Christmas.
My dad read the rules, set it up, and we were ready to play our first game (back then we learned by playing). My dad went first, got into a room and made his first accusation. He nailed it on the first attempt! (purely by chance). I remember thinking at the time, “This isn’t a very fun game” – I didn’t get to move, and there was a lot of setup for not much entertainment. We reset the game after this fluke and played many other times. On these later games we actually got to make a move ourselves, and to explore the nuances/strategy of the game for more than one guess; then it all made sense!
Over the years of playing, I’ve never encountered another first-guess win.

What are the chances?

What are the chances of nailing a first-guess? Well, in classic Cluedo, there are six suspects, six potential murder weapons, and nine possible crime locations. If we made a pure guess by picking, at random, one from each category, then there is a 1/6 × 1/6 × 1/9 = 1/324 chance of guessing the solution straight off ≈ 0.31%
However, this is not the complete story because you hold in your hand clue cards. These eliminate certain options. Your guess can be filtered by this information, increasing your chances.
The number of clue cards each player gets is dependent on the number of players in the game. The game can be played with between 3 – 6 players.

Who, What, Where?

There are 21 clue cards in game (6+6+9), and after one of each has been selected, this leaves (5+5+8=18) cards left to be distributed. Depending on the number of players, you could receive anywhere from three to six clue cards. Let’s look at the example where you receive six clue cards (as was the case when I played with my brother and my dad).
Depending on the distribution of the clue cards you receive, the odds change. For example, if you get two cards from each category, you have a 1/4 × 1/4 × 1/7 = 1/112 chance ≈ 0.89% of randomly guessing the solution, but if you receive no suspects, one weapon, and five locations, then your chances are 1/6 × 1/5 × 1/4 = 1/120 off ≈ 0.83%
Here’s a table of all the combinations:


As you can see, there is symmetry between the suspects and the weapons (which is obvious, as they both contain the same number of cards), and that your best chance of winning is having all of the cards from one of two these groups (essentially making that part of the guess a sure thing), and the extra card not being a room. This configuration increases your chances to 2.22%
When playing with three players, it's best to start off with all the suspects, or all the weapons, and your remaining card not being a room.
In a four player game, you might get five or four clue cards (I know, it's not fair to only get four!). If you get five cards, again, it's best to have either all the suspects, or all the weapons. If you get four cards to start, you want them all to be from either from the suspects or the weapons. The same for three cards.
If you get one of these 'pat' hands, if you have six cards, it's a 1/45 chance for a random guess. If you have five cards it's a 1/54 chance. If you have four cards, it's a 1/108 chance, and for three it's 1/162.
If you've never heard of the game before, you can buy it here.