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## Rare Snapoker Hands

Probability puzzles require you to weigh all the possibilities and pick the most likely outcome.

 Puzzle ID: #9476 Fun: (2.39) Difficulty: (2.52) Category: Probability Submitted By: snaps Corrected By: Winner4600

The card game Snapoker is a very complex game for four players, requiring the use of two decks of cards with all odd spades, even hearts, prime diamonds (including aces) and non-prime clubs (including aces) removed. Note that jacks are 11, queens are 12, and kings are 13. Two jokers are added to the deck and are each worth 15. The cards are dealt among the four players. A hand starts when the dealer shouts, "Let's Snapoker!" Following a frenetic minute of randomly throwing all cards onto the table, at the dealer's call of, "Heeeeeeere, piggy, piggy, piggy!" players are required to grab as many cards as they can. Once all cards have been grabbed, the total value of the cards in each player's hand is added up. Points (4, 3, 2 and 1) are awarded to players on each hand, with the highest card total receiving maximum points. In the event of a tie, the player who can drink a glass of milk through their nose the quickest will receive the greater points. A running score is kept throughout the game. The game is over when someone reaches a total of 518 points or when a fight breaks out, which is a common occurrence.

The other day I was playing Snapoker with three of the quickest-handed people I have ever known. In any hand I was lucky if I was able to grab three cards, while they grabbed the rest. On one hand I was able to grab four cards and, to my surprise, they were all the same number and same colour. What were the four cards I grabbed and what is the chance of obtaining such a four-card hand in Snapoker?

My hand could have been either four black 2's or four red 9's. All other numbers in a Snapoker deck have two black and two red cards, except for the two aces, which are both red, and the two jokers.

Now for the probability. In the case of black 2's, the odds of the first card being a black 2 would be 4/52. Given that, the odds of drawing another 2 would be 3/51 (then 2/50, then 1/49). Thus the probability of obtaining four black 2's is (4*3*2*1)/(52*51*50*49) which equates to a chance of 1 in 270,725. This probability then needs to be multiplied by 2 as four red 9's are also a possibility. Therefore, the probability of grabbing a four-card hand with the same number and colour is 2 in 270,725.

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