This post works with 5-card Poker hands drawn from a standard deck of 52 cards. The discussion is mostly mathematical, using the Poker hands to illustrate counting techniques and calculation of probabilities.

Working with poker hands is an excellent way to illustrate the counting techniques covered previously in this blog — multiplication principlepermutation and combination also covered here.

### How many different 5-card hands are possible in a standard deck of 52 cards?

There are 2, many possible 5-card Poker hands. Thus the probability of obtaining any one specific hand is 1 in 2, roughly 1 in 2. The probability of obtaining a given type of hands e. Thus this is primarily a counting exercise.

## Standard 52-card deck

Usually the order in which the cards are dealt is not important except in the case of stud poker. Thus the following three examples point to the same poker hand. The only difference is the order in which the cards are dealt.

The number of possible 5-card poker hands would then be the same as the number of 5-element subsets of 52 objects. The following is the total number of 5-card poker hands drawn from a standard deck of 52 cards. Other notations for areand.

Many calculators have a function for. Of course the calculation can also be done by definition by first calculating factorials. Thus the probability of obtaining a specific hand say, 2, 6, 10, K, A, all diamond would be 1 in 2, If 5 cards are randomly drawn, what is the probability of getting a 5-card hand consisting of all diamond cards?

It is. This is definitely a very rare event less than 0. The numerator 1, is the number of hands consisting of all diamond cards, which is obtained by the following calculation. The reasoning for the above calculation is that to draw a 5-card hand consisting of all diamond, we are drawing 5 cards from the 13 diamond cards and drawing zero cards from the other 39 cards.

Since there is only one way to draw nothingis the number of hands with all diamonds. If 5 cards are randomly drawn, what is the probability of getting a 5-card hand consisting of cards in one suit?

So we have the following derivation. Thus getting a hand with all cards in one suit is 4 times more likely than getting one with all diamond, but is still a rare event with about a 0. Some of the higher ranked poker hands are in one suit but with additional strict requirements. They will be further discussed below. Another example. What is the probability of obtaining a hand that has 3 diamonds and 2 hearts?

One theme that emerges is that the multiplication principle is behind the numerator of a poker hand probability. For example, we can think of the process to get a 5-card hand with 3 diamonds and 2 hearts in three steps. The first is to draw 3 cards from the 13 diamond cards, the second is to draw 2 cards from the 13 heart cards, and the third is to draw zero from the remaining 26 cards.

The third step can be omitted since the number of ways of choosing zero is 1.In many forms of pokerone is dealt 5 cards from a standard deck of 52 cards. The number of different 5 -card poker hands is. A wonderful exercise involves having students verify probabilities that appear in books relating to gambling. McGervey, one finds many interesting tables containing probabilities for poker and other games of chance. This article and the tables below assume the reader is familiar with the names for various poker hands.

I have done computations to verify McGervey's figures. This could be an excellent exercise for students who are studying probability.

There are 13 denominations A,K,Q,J,10,9,8,7,6,5,4,3,2 in the deck. One can think of J as 11, Q as 12, and K as Since an ace can be "high" or "low", it can be thought of as 14 or 1.

With this in mind, there are 10 five-card sequences of consecutive dominations. The following table displays computations to verify McGervey's numbers. There are, of coursemany other possible poker hand combinations. Those in the table are specifically listed in McGervey's book. The computations I have indicated in the table do yield values that are in agreement with those that appear in the book.

Straight flush 40 There are four suits spades, hearts, diamond, clubs. Choose 1 of 13 denominations to get four cards and combine with 1 card from the remaining Choose 1 denominaiton, pick 3 of 4 from it, choose a second denomination, pick 2 of 4 from it. Choose 1 suit, then choose 5 of the 13 cards in the suit. This figure includes all flushes. McGervey's figure does not include straight flushes listed above. Each denomination card can be 1 of 4 in the denomination. This figure includes all straights.

Choose 1 of 13 denominations, pick 3 of the four cards from it, then combine with 2 of the remaining 48 cards. This figure includes all full houses. McGervey's figure does not include full houses listed above. Choose 2 of the four aces, pick 1 card from remaining 48 and remove from consider other cards in that denominationchoose 1 card from remaining 44 and remove other cards from that denominationthen chose 1 card from the remaining The division by 3!

These are the same sets of three cards, just chosen in a different order. Choose 2 of the 4 threes, 2 of the 4 twos, and one card from the 44 cards that are not 2's or 3's.A playing card is a piece of specially prepared heavy paper, thin cardboard, plastic-coated paper, cotton-paper blend, or thin plastic, marked with distinguishing motifs and used as one of a set for playing card games.

Playing cards are typically palm-sized for convenient handling. A complete set of cards is called a pack UK English or deck US Englishand the subset of cards held at one time by a player during a game is commonly called a hand. A deck of cards may be used for playing a great variety of card games, with varying elements of skill and chance, some of which are played for money. Because playing cards are standardized and commonly available, they are used for other purposes, such as illusions, cartomancy, cardistry, and building card structures.

The front or "face" of each card carries markings that distinguish it from the other cards in the deck and determine its use under the rules of the game being played. The back of each card is identical for all cards in any particular deck, and usually of a single color or formalized design.

Usually every card will be smooth; however, some decks have braille to allow blind people to read the card number and suit. The backs of playing cards are sometimes used for advertising. The United States introduced the joker into the deck. The stylings of the joker and its function are almost identical to the Fool from the original French Tarot deck, which had been removed in the transformation to the standard card French deck.

It was devised for the game of Euchre, which spread from Europe to America beginning shortly after the American Revolutionary War and was very popular by the midth century.

In Euchre, the highest trump card is the Jack of the trump suit, called the right bower or bauer ; the second-highest trump, the left bower, is the Jack of the suit of the same color as trumps. The joker was invented c. The name of the card is believed to derive from juker, a variant name for Euchre. In the 19th century, a type of card known as a transformation playing card became popular in Europe and America.

In these cards, an artist incorporated the pips of the non-face cards into an artistic design. Popular legend holds that the composition of a deck of cards has religious, mystical, or astrological significance. Thus each suit of 13 cards represents the 13 months of the lunar year. Since the sidereal lunar month may be approximated to 28 days, each suit is equal to days of the year. Similarly, the whole deck of 52 cards represents the 52 weeks of the year.

Therefore, the whole deck is also equal to days of the year the positivist calendar. There are theories about whom the court cards represent. However, the Kings, Queens, and Jacks of Anglo-American designs today do not represent anyone in particular. They stem from designs produced in Rouen beforeand, by —67, these Rouen designs show well executed pictures in the court cards with the typical court costumes of the time.The standard card deck of French-suited playing cards is the most common pack of playing cards used today.

The most common pattern worldwide and the only pattern commonly available in Britain and America is the English pattern pack. Each suit includes an Acea KingQueen and Jackeach depicted alongside a symbol of its suit; and numerals or pip cards from the Deuce Two to the Ten, with each card depicting that many symbols pips of its suit.

Anywhere from one to six most often two or three since the midth century Jokersoften distinguishable with one being more colourful than the other, are added to commercial decks, as some card games require these extra cards.

The most popular standard pattern of the French deck is the English pattern [b] pictured belowsometimes referred to as the International pattern or Anglo-American pattern. Modern playing cards carry index labels on opposite corners or in all four corners to facilitate identifying the cards when they overlap and so that they appear identical for players on opposite sides.

For the court cards, this comprises the initial letter or letters from the name of that card. In other countries the letters may vary. Although French-suited, card packs are the most common playing cards used internationally, there are many countries or regions where the traditional pack size is only 36 Russia, Bavaria or 32 north and central Germany, Austria or where regional cards with smaller packs are preferred for many games.

For example, or card Italian-suited packs are common in Italy; and card Spanish-suited packs on the Iberian peninsula; and card German-suited packs are very common in Bavaria and Austria. In addition, tarot cards are required for games such as French tarot 78 cardswhich is widely played in France, and the Tarock family of games 42 or 54 cards played in countries like Austria and Hungary.

The English pattern pack originated in Britain which was importing French playing cards from Rouen and Antwerp by The earliest cards of the English pattern date to around But Britain only started manufacturing its own cards towards the end of the 16th century, when card production began in London.

These were based on the Rouen pattern, but unlike the traditional French cards, they dropped the names on the court cards. The English pattern evolved, in the process losing "some of its Rouen flavour and elegance and became more and more stylised. The figures took more space in the cards and many details were distorted. All early cards of this type were single-headed, but aroundthe double-headed cards, universally used on modern decks, appeared. Corner indices were added around During the 19th century, the English pattern spread all over the world and is now used almost everywhere, even in countries where traditional patterns and other suits are popular.

In America, the English pattern was copied onto wider cards. The fanciful design and manufacturer's logo commonly displayed on the ace of spades began under the reign of James I of Englandwho passed a law requiring an insignia on that card as proof of payment of a tax on local manufacture of cards.

Until August 4,decks of playing cards printed and sold in the United Kingdom were liable for taxable duty and the ace of spades carried an indication of the name of the printer and the fact that taxation had been paid on the cards. In the USA, standard playing cards are available in both "wide" and "narrow" sizes, referred to by one manufacturer as either 'poker' or 'bridge' sized; [5] nominal dimensions are summarized in the adjacent table.

However, there is no formal requirement for precise adherence and minor variations are produced by various manufacturers in different countries. The slightly narrower cards are more suitable for games such as bridge and some types of poker, where a number of cards must be held or concealed in a player's hand.

US casino shuffling machines have traditionally been designed for bridge-size narrow cards for these reasons. In other table games, such as 21 blackjacka modern casino may use hundreds or even thousands of decks per day, so paper cards are used for those, for economic reasons. Poker-size wide paper decks are used for 21 and other similar games. The thickness and weight of modern playing cards are subject to numerous variables related to their purpose of use and associated material design for durability, stiffness, texture and appearance.

Some decks include additional design elements. Casino blackjack decks may include markings intended for a machine to check the ranks of cards, or shifts in rank location to allow a manual check via an inlaid mirror. Many casino decks and solitaire decks have four indices instead of just two. Some modern decks have bar code markings on the edge of the face to enable them to be sorted by machine for playing duplicate bridge, especially simultaneous events where the same hands may be played at many different venues.

Some decks have large indices for clarity. These are sometimes sold as 'seniors' cards' for older people with limited eyesight, but may also be used in games like stud pokerwhere being able to read cards from a distance is a benefit and hand sizes are small. The standard French-suited pack uses black for the spades and clubs, and red for the hearts and diamonds.

However, some packs use four colours for the suits in order to make it easier to tell them apart. When giving the full written name of a specific card, the rank is given first followed by the suit, e.I assume it is not. So the number of possibilities is C 12,5. Note: The notation NCk is used to symbolize the binomial coefficient with parameters N, k, calculated as N!

There are 12 face cards in the deck K, Q, J of all four suits. There is only one way to choose all 4 kings. The other card can be any of the remaining 48 cards in the deck, so there are 48 ways to choose 4 kings. There are 4 ways to choose an Ace. Same goes for diamonds. This makes a total of:. There are 39 non-spade cards in the deck, and 39C5 ways to choose a hand containing no spades. All the other hands in the 52C5 possible hands contain at least 1 spade.

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Beware of appropriation posing as a costume. How 'secretly in love' bandmates finally came out. The undocumented Latina leading a massive movement. Wallace grills Lara Trump for breaking debate rule. Colleges face existential crises waiting for stimulus. North Korea unveils new weapons at military parade.The coefficient for female is the log of odds ratio between the female group and male group: log(1.

So we can get the odds ratio by exponentiating the coefficient for female. Most statistical packages display both the raw regression coefficients and the exponentiated coefficients for logistic regression models. The table below is created by Stata. In other words, the odds of being in an honors class when the math score is zero is exp(-9.

These odds are very low, but if we look at the distribution of the variable math, we will see that no one in the sample has math score lower than 30.

In fact, all the test scores in the data set were standardized around mean of 50 and standard deviation of 10. So the intercept in this model corresponds to the log odds of being in an honors class when math is at the hypothetical value of zero. How do we interpret the coefficient for math. We will use 54. Then the conditional logit of being in an honors class when the math score is held at 54 isWe can examine the effect of a one-unit increase in math score.

When the math score is held at 55, the conditional logit of being in an honors class isWe can say now that the coefficient for math is the difference in the log odds.

In other words, for a one-unit increase in the math score, the expected change in log odds is. Can we translate this change in log odds to the change in odds.

Recall that logarithm converts multiplication and division to addition and subtraction.

**5-card Poker STRAIGHT Probability and Odds**

Its inverse, the exponentiation converts addition and subtraction back to multiplication and division. Applying such a model to our example dataset, each estimated coefficient is the expected change in the log odds of being in an honors class for a unit increase in the corresponding predictor variable holding the other predictor variables constant at certain value. Each exponentiated coefficient is the ratio of two odds, or the change in odds in the multiplicative scale for a unit increase in the corresponding predictor variable holding other variables at certain value.

Here is an example. In all the previous examples, we have said that the regression coefficient of a variable corresponds to the change in log odds and its exponentiated form corresponds to the odds ratio.

This is only true when our model does not have any interaction terms. The interpretation of the regression coefficients become more involved. In this simple example where we examine the interaction of a binary variable and a continuous variable, we can think that we actually have two equations: one for males and one for females.Petco does a great job of incorporating a seasonal element into their cover photo 7.

### Find the number of possible 5-card hands that contain the cards specified.?

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