The Pao De Kuai (PDK) Card Game Goal is to Get Rid of all the Cards: Probability Assignment, NUS, Singapore

Assignment Brief: Probability calculations for card game

The Pao De Kuai (PDK) card game goal is to get rid of all the cards in your hands

The PDK deck is the following

Heart: 3 4 5 6 7 8 9 10 J Q K A
Spade: 3 4 5 6 7 8 9 10 J Q K A
Diamond: 3 4 5 6 7 8 9 10 J Q K A
Club: 3 4 5 6 7 8 9 10 J Q K 2

As you can see the deck contains only 48 cards (only three aces and one 2)

All the cards are dealt to 3 players so each player received 16 cards

Rule: to get rid of the cards you need to play the right combination and with a higher rank than the combination of the previous player.

My objective

I want to calculate the probability for a player to have a specific combination using combinatorics

This probability depends on the number of cards in the hand and the number of cards remaining in the other hands as well.

The different possible combinations are

single

Ex: 4

N Connected singles (straight) – must be at least 5 connected cards
N is the number of connected 5must be at least 5

Ex: 4 5 6 7 8 9 10

Pair

Ex: 4 4

N Connected pairs
N is the number of connected pairs, so must be at least 2

Ex: 4 4 5 5 6 6

4 of a kind (called ‘bomb’)

Ex: 4 4 4 4

4-3 (4 of a kind + 3 cards of your choice from your hand)

Ex: 4 4 4 4 8 J K

N Connected 4-3 (the 4 of a kind must be connected)

Ex: 4 4 4 4 5 5 5 5 6 6 7 8 9 10

3-2 (3 of a kind + 2 cards of your choice form your hand

Ex: 4 4 4 J A

N Connected 3-2 (the 3 of a kind must be connected)

Ex: 4 4 4 5 5 5 6 8 9 9

Questions:

Probability for a player to have:

At least one pair
At least one 3-2
At least one 4-3
At least one 4 of a kind
At least 2 connected-pairs
At least 3 connected pairs
At least 4 connected pairs
At least 5 connected pairs
At least 6 connected pairs
At least 7 connected pairs
At least 8 connected pairs
At least 1 4-3
At least 2 4-3
At least 1 3-2
At least 2 3-2
At least 3 3-2
At least 1 straight of 5 (connected-singles)
At least 1 straight of 6
…
At least 1 straight of 13