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## Lucky Cards

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

 Puzzle ID: #32515 Fun: (2.71) Difficulty: (2.36) Category: Probability Submitted By: Gizzer Corrected By: Radioguy

After an intense week of play-downs, the National Draw Poker Championship has come down to the final hand between two fierce competitors: Tex Holden, the defending champion; and Rae Sonheim, the unknown of the tournament, who has amazed the spectators with her cool bluffs. In what is the first of a number of amazing coincidences of the night, both players have an equal stake.

Although Tex kept his poker face intact, his excitement mounted as he examined each card he was dealt. The first card he received was his lucky card, the King of Clubs, which he considered a very good omen. Perhaps it was, as each of the next four cards were of the same value as each other, giving Tex one of the best possible hands: four of a kind.

Meanwhile, across the table, Rae waited until all the cards were dealt before looking at her cards. She, too, was pleasantly surprised as she had received four of a kind as well, along with the Queen of Spades, which is her lucky card.

Each player, believing he or she had an unbeatable hand, prepared to end the tournament then and there with a careful strategy designed to win all the chips.

After many rounds of raises and counter-raises, at long last, all the chips were riding on this one hand. The better hand would take the pot, and the trophy.

Tex, having had his last raise called, gleefully displayed his hand first. All eyes turned to Rae, but, as usual her face showed no emotion.

Before the outcome is revealed, can you determine the probability that Tex has retained his title?

As a single King and Queen have been identified in the hands, neither player can have 4 Kings or 4 Queens. That leaves 11 possibilities (in decreasing value):

4 Aces
4 Jacks
4 Tens
4 Nines
4 Eights
4 Sevens
4 Sixes
4 Fives
4 Fours
4 Threes
4 Twos

The probability that Tex has any of these hands is the same 1/11.

If Tex has 4 Aces, Rae's 4 of a kind cannot outrank his, so the probability that he would win the hand is 1.

If Tex has 4 Jacks, then of the remaining 10 possible hands that Rae holds, only 1 can beat him. Therefore the probability that Tex would win with 4 Jacks is 9/10.

With 4 Tens, the probability that Tex would win is 8/10, and so on.

If Tex has 4 Twos, any four of kind held by Rae would win, so the probability of Tex winning would be 0.

Therefore, the probability that Tex will win can be expressed as:

P = 1/11(10/10 + 9/10 + 8/10 +7/10 + 6/10 + 5/10 + 4/10 + 3/10 + 2/10 + 1/10 + 0/10)

P = 55/110

P = 0.5

Tex's poker face finally fell when he saw Rae smile sweetly and slowly fan out her 4 Eights, which beat his 4 Sixes. He quickly recovered his composure, and congratulated Rae on a hand well played.

Of course, Tex no longer considers the King of Clubs to be his lucky card.

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