Detailed and step by step solution please. 50 points In a five-card poker hand,

Question

Detailed and step by step solution please. 50 points In a five-card poker hand, a flush is defined as five cards all of the same suit, regardless of rank, and a straight is defined as five cards in sequential order, say 8, 9, 10, J ack, Queen, dless of suit. One can show that a straight is more likely than a flush, making the flush more valuable in five-card poker. If we play a four-card poker game, the hands are defined in the same way (a flush is four cards of the same suit, a straight is four cards in sequential order). Dr. Coe wants to play a four-card game of poker, and his deck of cards is missing all of the 2s, 3s, and 4s (it has only 40 cards). What is the probability of a flush and the probability of a straight for this four-card game from his 40 card deck? Which hand is more valuable?

Explanation / Answer

Please note nCr = n! / [(n-r)!*r!] and Probability = Favourable outcomes/Total Outcomes

A hand which occurs lesser number of times is more valuable.

This deck contains 40 cards, with 10 cards of each suit, from 5,6,7,8,9.10,J,Q.K and A.

______________________________________________________________________

Probability of getting a flush:

Favourable outcomes = Getting a flush = choosing 1 suit out of the 4 and then choosing 4 cards out of the 12 = 4C1 x 12C4 = 4 x 495 = 1,980 ways of getting a flush.

Total Outcomes = Choosing any 4 cards out of the 40 = 40C4 = 91,390

Required Probability = 1,980/91,390 = 0.021665

________________________________________________________________________

Probability of any straight: In a straight ( a sequence) we start with a 5, eg (5 6 7 8),Other options are starting with a 6 , 7 , 8 , 9, 10 or J. So there are 7 possibilities with which we can start a sequence, hence there are 7 possible straight sequences. After we choose a starting card, we can choose 1 type of card from 4 (clubs diamonds hearts or spades) for each of the 4 cards.

The total number of straights = 7C1 x 4C1 x4C1 x4C1 x4C1 = 7 x 4 x 4 x 4 x 4 = 1,792

Total Outcomes = Choosing any 4 cards out of the 40 = 40C4 = 91,390

Required Probability = 1,792/91,390 = 0.0196

_______________________________________________________________________

So the hand which occurs the lesser number of times amongst the 2 is more valuable. The straight(1792) occurs lesser than the flush(1980) in this game, and hence the straight is more valuable in th e40 cards game.

Related Questions

Navigate

• Browse (All)
• Browse D
• Subjects

---

---

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.