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Chapter 2: Problem 6

A poker deck consists of fifty-two cards, representing thirteen denominations ( 2 through ace) and four suits (diamonds, hearts, clubs, and spades). A five- card hand is called a flush if all five cards are in the same suit but not all five denominations are consecutive. Pictured below is a flush in hearts. Let \(N\) be the set of five cards in hearts that are not flushes. How many outcomes are in N? [Note: In poker, the denominations \((\mathrm{A}, 2,3,4,5)\) are considered to be consecutive (in addition to sequences such as \((8,9\), \(10, \mathrm{~J}, \mathrm{Q}))\).]

Short Answer

Expert verified
The number of outcomes in N is 1276.

Step by step solution

01

Total Possible Combinations

To get the total possible combinations of drawing 5 cards of hearts out of 13, we need to use combinatorics. The formula for combination is \(C(n, r) = n! / [r!(n-r)!]\) where n is the total number of options, r is the number of options chosen at a time, and '!' denotes factorial. For our case n=13 (as there are 13 cards in one suit) and r=5 (as we are choosing 5 cards) so, using the formula, we get \(C(13, 5) = 1287\). This gives us the total number of five-card combinations in hearts.
02

Flushes or Consecutive Denominations

Next, we need to consider how many of these combinations involve five consecutive denominations, which would be considered flushes. According to the exercise, it should be noted that A, 2, 3, 4, 5, and the sequence 9,10,J,Q, and A are counted as sequences, thus they have to be added. There are 9 sets of five consecutives in a 13-card suit. Adding the 2 special sequences A, 2, 3, 4, 5 and 9, 10, J, Q, A, we have a total of 11 flushes.
03

Calculate the Outcome

Finally, to get outcomes in \(N\), which are the five-card hands that are not flushes, we subtract the number of flushes from the total possible combinations. So, we get \(N = 1287 - 11 = 1276\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Poker Deck
A poker deck is a standard set of playing cards that consists of 52 distinct cards. These cards are divided into four suits: diamonds, hearts, clubs, and spades. Each suit contains 13 cards, ranging from the number 2 up to the Ace.

Here are the ranks of the cards you will find in each suit:
  • Number cards: 2 through 10
  • Face cards: Jack (J), Queen (Q), King (K)
  • Ace (A), which can serve both as the lowest or highest card depending on the game context
The poker deck is fundamental to many card games, and understanding its composition is key to grasping various strategies and outcomes in games like poker.
Five-Card Hand
In poker, a five-card hand refers to a specific collection of five cards dealt from the deck. These hands are used to determine winners in most versions of poker and have different rankings depending on the combinations, such as pairs, straights, flushes, etc.

To illustrate:
  • A hand can be as simple as a single pair, like two cards of the same rank.
  • It can be as complex as a royal flush, which is a hand consisting of the ten, jack, queen, king, and ace, all from the same suit.
Understanding the different types of five-card hands is crucial because the value of the hand can determine the outcome of a poker game.
Flush
A flush is one of the many types of poker hands. It consists of any five cards of the same suit, but the specific denominations don't follow any particular sequence. This means that while suits match, the actual numbers or ranks on the cards do not need to be in order.

Here鈥檚 what to keep in mind:
  • A flush can include cards like 3, 6, 9, J, Q as long as they are all from the same suit.
  • What distinguishes a flush from other poker hands is its emphasis on suit uniformity rather than rank order.
Flushes are a basic yet important concept in poker, as they determine a hand's strength relative to others.
Consecutive Denominations
Consecutive denominations in poker refer to a sequence where the ranks of the cards follow each other in a sequence within a suit. This can form a straight or contribute to a straight flush if they are also of the same suit.

Examples of consecutive denominations include:
  • A, 2, 3, 4, 5
  • 9, 10, J, Q, K
In some cases, denominations can wrap around, like from A to 2. However, note that these sequences alter the type of poker hand. Consecutive denominations offer a structured take on card order, providing valuable context for creating specific poker hands.

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Most popular questions from this chapter

A fair coin is tossed four times. What is the probability that the number of heads appearing on the first two tosses is equal to the number of heads appearing on the second two tosses?

Two sections of a senior probability course are being taught. From what she has heard about the two instructors listed, Francesca estimates that her chances of passing the course are \(0.85\) if she gets Professor \(X\) and \(0.60\) if she gets Professor \(Y\). The section into which she is put is determined by the registrar. Suppose that her chances of being assigned to Professor \(X\) are four out of ten. Fifteen weeks later we learn that Francesca did, indeed, pass the course. What is the probability she was enrolled in Professor \(X\) 's section?

The governor of a certain state has decided to come out strongly for prison reform and is preparing a new early release program. Its guidelines are simple: prisoners related to members of the governor's staff would have a \(90 \%\) chance of being released early; the probability of early release for inmates not related to the governor's staff would be \(0.01\). Suppose that \(40 \%\) of all inmates are related to someone on the governor's staff. What is the probability that a prisoner selected at random would be eligible for early release?

School board officials are debating whether to require all high school seniors to take a proficiency exam before graduating. A student passing all three parts (mathematics, language skills, and general knowledge) would be awarded a diploma; otherwise, he or she would receive only a certificate of attendance. A practice test given to this year's ninety-five hundred seniors resulted in the following numbers of failures: $$ \begin{array}{lc} \hline \text { Subject Area } & \text { Number of Students Failing } \\ \hline \text { Mathematics } & 3325 \\ \text { Language skills } & 1900 \\ \text { Ceneral knowledge } & 1425 \\ \hline \end{array} $$ If "Student fails mathematics," "Student fails language skills," and "Student fails general knowledge" are independent events, what proportion of next year's seniors can be expected to fail to qualify for a diploma? Does independence seem a reasonable assumption in this situation?

Four cards are dealt from a standard fifty-two-card poker deck. What is the probability that all four are aces given that at least three are aces? (Note: There are 270,725 different sets of four cards that can be dealt. Assume that the probability associated with each of those hands is \(1 / 270,725 .)\)
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