I am trying to calculate the probability that each card is inside the envelope as the information revealed progresses during the game.
The deck is composed of 21 cards, which are divided into three different classes ('Suspects', 'Weapons' and 'Rooms'). The first two classes contain 6 cards each, while the third one contains 9 cards. Then one card is removed for each class and placed in an enevelope, so that the deck now contains 18 cards in total. After shuffling, these 18 cards are dealt among 6 players (I am insterested in this case only), so that each player holds 3 cards (not necessarily one for each class).
Let's assume I am one of the 6 players. At this point, the probability that any of the cards belonging to the class 'Rooms' has been removed from the deck and placed in the envelope is 1/9, while for the other two classes is 1/6. Conversely the probabilities that these kind of cards are in the hands of the players are respectively 8/9 and 5/6.
Then let's assume that I become aware of the three cards that I have been dealt with. At this point I want to answer these two kinds of questions:
- What is, for example, the probability that the card 'Kitchen' (which belongs to the 'Rooms' class) is in the enevelope conditioned to the fact that it is not among my three cards?
- What is, for example, the probability that the card 'Kitchen' is in the envelope conditioned to the fact that it is not among my cards but 'Living Room' (another card from the class 'Rooms') is?
I know that we are dealing with conditional probabilities, but I cannot figure out how to calculate them, as the initial probability of the card 'Kitchen' to be removed from the deck and placed in the envelope is 1/9, which is different from the probability that this card will end in the hands of each player (which if I am not wrong should be 4/27).
There is $1$ card of Rooms class in envelope and $8$ cards among players.
i) Now if you have no card of Rooms class in your hand, there are still $8$ Rooms cards out there among other players. So the probability still remains the same that the Kitchen card is in the envelope is $\frac{1}{9}$ and in hands of players is $\frac{8}{9}$.
ii) If you have one card of Rooms class in your hand but it is not Kitchen card. Now this information does change the probability (for you) that the Kitchen card is in the envelope as it is now one of the $8$ cards. That is simply $\frac{1}{8}$.
If you want to write it as conditional probability -
Event $A$ is that Kitchen card is in the envelope. Event $B$ is that it is one of the $8$ cards distributed (including one in envelope),
$P(B) = \displaystyle \frac{8 \choose 7}{9 \choose 8} = \frac{8}{9}$.
$P(A \cap B) = \frac{1}{9}$. $P(A|B) = \frac{1/9}{8/9} = \frac{1}{8}$.
Similarly,
iii) If you have two cards of Rooms class and none of them are Kitchen card, the probability of Kitchen card in envelope is $\frac{1}{7}$.
iv) If you have three non-Kitchen cards from Rooms class, the probability of Kitchen card in envelope is $\frac{1}{6}$.