I am trying to find the probability of drawing a heart as the dealer in this game called Pedro. I have been working on this problem for quite some time and I'm stumped. Basically this is how the play works.
You deal out 36 cards to 4 players so each player has 9 cards and then one round of bidding starts. After the bidding is done the bidder determines the trump or high suit.
Everyone then discards all of their cards that are not trump or the suit specified.
The dealer then asks each player how many cards they need in order to get to 6 total cards in their hand.
After passing out the cards to each player, including themselves, the dealer then gets to look at the left over cards which can very from 0 to 6 cards.
Note that in this game the five of the same color but different suit counts as a trump card so instead of 13 for one suit it is now 14. Ex: Trump suit is hearts, the 5 of diamonds is now a trump.
Note if a player has 7,8, or 9 of the same suit in their hand then they keep these cards and just do not take any extra cards from the dealer. That is the complicated portion of calculating the probability of the dealer picking up an extra Trump card in the extra cards at the end. Here are some examples of the play.
There are 16 cards left for the dealer to deal out after the trump suit has been determined.
Player 1 has 2 trump;
Player 2 has 3 trump;
Player 3 has 3 trump;
Dealer has 2 trump;
Currently there is 10 Trump out and 4 trump left to possibly get.
The dealer gives Player 1 four cards Then Player 2 three cards Then player 3 three cards Finally gives themselves four cards.
This totals to 14 cards passed out and thus 2 cards left over.
The dealer looks at these cards and 1 is trump and he keeps that card and discards another card from his hand that is not trump.
Basically, what is the probability in all scenarios that the dealer picks up at least 1 extra trump from the extra cards?