Pinochle is a card game in which the only cards are, in decreasing rank order A,
ID: 3254698 • Letter: P
Question
Pinochle is a card game in which the only cards are, in decreasing rank order A,10,K,Q,J and 9. There are no other cards. However, there are two identical copies of each card, i.e., 2 Aces of Hearts, 2 Aces of Clubs, 2 Aces of Spades and 2 Aces of Diamonds, i.e, a total of 8 of each rank for a total of 48 cards. Pinochle is played by 4 players, each of whom has a 12 card hand. Give the count and probability of each hand:
1.) "100 Aces": 4 Aces of different suits—in this example, we want no other Aces in our 12 card hand
2.)"800 Kings": All 8 Kings
3.) "Flush": 12 cards of the same suit
4.) "Double Pinochle": both Jacks of Diamonds and both Queens of spades in a 12 card hand
5.) "Double Marriage": Both kings and queens of one suit
Explanation / Answer
12 cards hand can be made from 48 cards in 48C12 ways
(1) 100 Aces
4 different aces can be selected in 2C1*2C1*2C1*2C1 = 2^4 = 16 ways
Remaining 8 cards can be selected from 40 cards in 40C8 ways
hence total number of possible hands with 4 aces of different suits = 40C8 * 16
Probability of 100 Aces = (40C8 * 16)/(48C12) = 0.0177
(2) 800 kings
All 8 kings can be selected in 1 way
Remaining 4 cards can be selected in 40C4 ways
probability of 800 kings = (40C4)/(48C12) = 0.00000131
(3) Flush
12 cards of the same suit can be selected in 4 ways
probability of flush = 1/(48C12)
(4)Double Pinochle
Both jacks of diamonds and both queens of spaces can be selected in only 1 ways
Remaining 8 cards can be selected from 44 cards in 44C8 ways
Probability of Double pinochle = 44C8/48C12 = 0.00254
(5) Double marriage
Both Kings and queens of one suit can be selected in 4C1 = 4 ways
Remaining 8 cards can be selected from 44 cards in 44C8 ways
Probability of Double pinochle = 44C8*4/48C12 = 0.0102