Find the probability of drawing a heart and a spade in standard deck of 52 cards
ProbabilityProbability is the study of experiments. Experiments result in outcomes (also called simple events). A set or collection of outcomes is called an event. These concepts can be illustrated with a deck of cards. Pulling a card from the deck is an experiment. Getting the ace of spades is an outcome (or simple event) (there are 52 possible outcomes for the experiment of drawing a card from a deck. Events include getting a spade (which event contains 13 outcomes), or getting a king (which event contains 4 outcomes).Probabilities are assigned to the outcomes of an experiment. We shall only consider experiments where all the outcomes are equally likely. Hence for drawing a card from a deck, each outcome has probability 1/52. The probability of an event is the sum of the probabilities of the outcomes in the event, hence the probability of drawing a spade is 13/52 = 1/4, and the probability of drawing a king is 4/52 = 1/13. Although the fact that the probability of getting a spade is 1/4 may make it seem that events are equally likely (since there are four suits), the probability of getting a face card is not equal to the probability of getting a number card. The only way to calculate the probability of an event is to sum the probabilities of the (equally likely) outcomes in the event. However, identifying what the equally likely outcomes are can be subtle as the following exercise illustrates. Show
Exercise: What are the equally likely outcomes for flipping a pair of coins? What are the equally likely outcomes for rolling a pair of dice? Additive ruleSince the the probability of an event is the sum of the probabilities of the outcomes which comprise the event, one might assume that the probability of an event is the sum of the probabilities of any events which comprise that event. However, The probability of getting a black card or an ace [which we may denote as P(black or ace)] is not P(black) + P(ace) since the former is 28/52 (there are 26 black cards and 2 red aces) while the latter is 26/52 + 4/52. The discrepancy is due to the fact that the black aces are counted twice on the right hand side, once with the black cards and once with the aces. Correcting for the double counting provides the additive rule for arbitrary events: P(A or B) = P(A) + P(B) - P(A and B). Indeed 28/52 = 26/52 + 4/52 - 2/52 (there are two black aces).Definition. Two events A and B are said to be mutually exclusive (or disjoint) if their intersection is empty (or equivalently, P(A and B) = 0). For example, the events getting a club and getting a one-eyed jack are mutually excllusive because the one-eyed jacks are the jacks of spades and hearts. The events getting a heart and getting a one-eyed jack are not mutually exclusive, because the jack of hearts is both a heart and a one eyed-jack. The additve rule for mutually exclusive events is P(A or B) = P(A) + P(B) (because P(a and B) which should be subtracted from the right hand side is equal to zero). Definiton: A' (read A complement) is the set of outcomes which are not in A. It follows that P(A or A') = 1 and P(A and A') = 0. Competencies:WHat is P(black or 2) What is P(Black and 2)? If you fip 3 coins, what is the probability of getting two heads? If you roll two dice, what is the probability that the sum of the pips will be 5? If P(A)=.4, P(B)=.3 and P(A and B)=.2, what is P(A or B)=? If A is the set of one-eyed jacks, what is A'? Reflection: Can you think of an experiment where the outcomes are not equally likely events? Challenge: May 2002 return to index
A standard deck contains an equal number of hearts, diamonds, clubs, and spades. So the probability of drawing a heart is $\frac{1}{4}$. There are four 7s in a standard deck, and there are a total of 52 cards. So the probability of drawing $\mathbf{a} 7$ is $\frac{1}{13}$. Nội dung chính Show
In playing cards, a suit is one of the categories into which the cards of a deck are divided. Most often, each card bears one of several pips (symbols) showing to which suit it belongs; the suit may alternatively or additionally be indicated by the color printed on the card. The rank for each card is determined by the number of pips on it, except on face cards. Ranking indicates which cards within a suit are better, higher or more valuable than others, whereas there is no order between the suits unless defined in the rules of a specific card game. In a single deck, there is exactly one card of any given rank in any given suit. A deck may include special cards that belong to no suit, often called jokers. History[edit]Modern Western playing cards are generally divided into two or three general suit-systems. The older Latin suits are subdivided into the Italian and Spanish suit-systems. The younger Germanic suits are subdivided into the German and Swiss suit-systems. The French suits are a derivative of the German suits but are generally considered a separate system.[1][2] Origin and development of the Latin suits[edit]
The earliest card games were trick-taking games and the invention of suits increased the level of strategy and depth in these games. A card of one suit cannot beat a card from another regardless of its rank. The concept of suits predate playing cards and can be found in Chinese dice and domino games such as Tien Gow. Chinese money-suited cards are believed to be the oldest ancestor to the Latin suit-system. The money-suit system is based on denominations of currency: Coins, Strings of Coins, Myriads of Strings (or of coins), and Tens of Myriads. Old Chinese coins had holes in the middle to allow them to be strung together. A string of coins could easily be misinterpreted as a stick to those unfamiliar with them. By then the Islamic world had spread into Central Asia and had contacted China, and had adopted playing cards. The Muslims renamed the suit of myriads as cups; this may have been due to seeing a Chinese character for "myriad" (万) upside-down. The Chinese numeral character for Ten (十) on the Tens of Myriads suit may have inspired the Muslim suit of swords.[3] Another clue linking these Chinese, Muslim, and European cards are the ranking of certain suits. In many early Chinese games like Madiao, the suit of coins was in reverse order so that the lower ones beat the higher ones. In the Indo-Persian game of Ganjifa, half the suits were also inverted, including a suit of coins. This was also true for the European games of Tarot and Ombre. The inverting of suits had no purpose in terms of play but was an artifact from the earliest games. These Turko-Arabic cards, called Kanjifa, used the suits coins, clubs, cups, and swords, but the clubs represented polo sticks; Europeans changed that suit, as polo was an obscure sport to them. The Latin suits are coins, clubs, cups, and swords. They are the earliest suit-system in Europe, and were adopted from the cards imported from Mamluk Egypt and Moorish Granada in the 1370s. There are four types of Latin suits: Italian, Spanish, Portuguese,[c] and an extinct archaic type.[4][5] The systems can be distinguished by the pips of their long suits: swords and clubs.
Despite a long history of trade with China, Japan was not introduced to playing cards until the arrival of the Portuguese in the 1540s. Early locally made cards, Karuta, were very similar to Portuguese decks. Increasing restrictions by the Tokugawa shogunate on gambling, card playing, and general foreign influence, resulted in the Hanafuda card deck that today is used most often for fishing-type games. The role of rank and suit in organizing cards became switched, so the hanafuda deck has 12 suits, each representing a month of the year, and each suit has 4 cards, most often two normal, one Ribbon and one Special (though August, November and December each differ uniquely from this convention). Invention of German and French suits[edit]
During the 15th-century, manufacturers in German speaking lands experimented with various new suit systems to replace the Latin suits. One early deck had five suits, the Latin ones with an extra suit of shields.[9] The Swiss-Germans developed their own suits of shields, roses, acorns, and bells around 1450.[10] Instead of roses and shields, the Germans settled with hearts and leaves around 1460. The French derived their suits of trèfles (clovers or clubs ♣), carreaux (tiles or diamonds ♦), cœurs (hearts ♥), and piques (pikes or spades ♠) from the German suits around 1480. French suits correspond closely with German suits with the exception of the tiles with the bells but there is one early French deck that had crescents instead of tiles. The English names for the French suits of clubs and spades may simply have been carried over from the older Latin suits.[11] Tarot[edit]Beginning around 1440 in northern Italy, some decks started to include an extra suit of (usually) 21 numbered cards known as trionfi or trumps, to play tarot card games.[12] Always included in tarot decks is one card, the Fool or Excuse, which may be part of the trump suit depending on the game or region. These cards do not have pips or face cards like the other suits. Most tarot decks used for games come with French suits but Italian suits are still used in Piedmont, Bologna, and pockets of Switzerland. A few Sicilian towns use the Portuguese-suited Tarocco Siciliano, the only deck of its kind left in Europe. The esoteric use of Tarot packs emerged in France in the late 18th century, since when special packs intended for divination have been produced. These typically have the suits cups, pentacles (based on the suit of coins), wands (based on the suit of batons), and swords. The trump cards and Fool of traditional card playing packs were named the Major Arcana; the remaining cards, often embellished with occult images, were the Minor Arcana. Neither term is recognised by card players.[13][14] Suits in games with traditional decks[edit]Trumps[edit]In a large and popular category of trick-taking games, one suit may be designated in each deal to be trump and all cards of the trump suit rank above all non-trump cards, and automatically prevail over them, losing only to a higher trump if one is played to the same trick.[15] Non-trump suits are called plain suits.[16] Special suits[edit]Some games treat one or more suits as being special or different from the others. A simple example is Spades, which uses spades as a permanent trump suit. A less simple example is Hearts, which is a kind of point trick game in which the object is to avoid taking tricks containing hearts. With typical rules for Hearts (rules vary slightly) the queen of spades and the two of clubs (sometimes also the jack of diamonds) have special effects, with the result that all four suits have different strategic value. Tarot decks have a dedicated trump suit. Chosen suits[edit]Games of the Karnöffel Group have between one and four chosen suits, sometimes called selected suits or, misleadingly, trump suits. The chosen suits are typified by having a disrupted ranking and cards with varying privileges which may range from full to none and which may depend on the order they are played to the trick. For example, chosen Sevens may be unbeatable when led, but otherwise worthless. In Swedish Bräus some cards are even unplayable. In games where the number of chosen suits is less than four, the others are called unchosen suits and usually rank in their natural order. Ranking of suits[edit]Whist-style rules generally preclude the necessity of determining which of two cards of different suits has higher rank, because a card played on a card of a different suit either automatically wins or automatically loses depending on whether the new card is a trump. However, some card games also need to define relative suit rank. An example of this is in auction games such as bridge, where if one player wishes to bid to make some number of heart tricks and another to make the same number of diamond tricks, there must be a mechanism to determine which takes precedence in the bidding order. There is no standard order for the four suits and so there are differing conventions among games that need a suit hierarchy. Examples of suit order are (from highest to lowest):
Pairing or ignoring suits[edit]The pairing of suits is a vestigial remnant of Ganjifa, a game where half the suits were in reverse order, the lower cards beating the higher. In Ganjifa, progressive suits were called "strong" while inverted suits were called "weak". In Latin decks, the traditional division is between the long suits of swords and clubs and the round suits of cups and coins. This pairing can be seen in Ombre and Tarot card games. German and Swiss suits lack pairing but French suits maintained them and this can be seen in the game of Spoil Five.[17] In some games, such as blackjack, suits are ignored. In other games, such as Canasta, only the color (red or black) is relevant. In yet others, such as bridge, each of the suit pairings are distinguished. In contract bridge, there are three ways to divide four suits into pairs: by color, by rank and by shape resulting in six possible suit combinations.
Four-color suits[edit]Some decks, while using the French suits, give each suit a different color to make the suits more distinct from each other. In bridge, such decks are known as no-revoke decks, and the most common colors are black spades, red hearts, blue diamonds and green clubs, although in the past the diamond suit usually appeared in a golden yellow-orange. A pack occasionally used in Germany uses green spades (comparable to leaves), red hearts, yellow diamonds (comparable to bells) and black clubs (comparable to acorns). This is a compromise deck devised to allow players from East Germany (who used German suits) and West Germany (who adopted the French suits) to be comfortable with the same deck when playing tournament Skat after the German reunification.[18] Other suited decks[edit]Suited-and-ranked decks[edit]A large number of games are based around a deck in which each card has a rank and a suit (usually represented by a color), and for each suit there is exactly one card having each rank, though in many cases the deck has various special cards as well. Examples include Mü und Mehr, Lost Cities, DUO, Sticheln, Rage, Schotten Totten, UNO, Phase 10, Oh-No!, Skip-Bo, Roodles, and Rook. Other modern decks[edit]Decks for some games are divided into suits, but otherwise bear little relation to traditional games. An example would be the board game Taj Mahal, in which each card has one of four background colors, the rule being that all the cards played by a single player in a single round must be the same color. The selection of cards in the deck of each color is approximately the same and the player's choice of which color to use is guided by the contents of their particular hand. In the trick-taking card game Flaschenteufel ("The Bottle Imp"), all cards are part of a single sequence ranked from 1 to 37 but split into three suits depending on its rank. players must follow the suit led, but if they are void in that suit they may play a card of another suit and this can still win the trick if its rank is high enough. For this reason every card in the deck has a different number to prevent ties. A further strategic element is introduced since one suit contains mostly low-ranking cards and another, mostly high-ranking cards. Whereas cards in a traditional deck have two classifications—suit and rank—and each combination is represented by one card, giving for example 4 suits × 13 ranks = 52 cards, each card in a Set deck has four classifications each into one of three categories, giving a total of 3 × 3 × 3 × 3 = 81 cards. Any one of these four classifications could be considered a suit, but this is not really enlightening in terms of the structure of the game. Uses of playing card suit symbols[edit]Card suit symbols occur in places outside card playing:
Character encodings[edit]In computer and other digital media, suit symbols can be represented with character encoding, notably in the ISO and Unicode standards, or with Web standard (SGML's named entity syntax):
Unicode is the most frequently used encoding standard, and suits are in the Miscellaneous Symbols Block (2600–26FF) of the Unicode. In some card games the card suits have a dominance order, for example: club (lowest) - diamond - heart - spade (highest). That led to in spades being used to mean more than expected, in abundance, very much.[20] In European games, the order is often different: diamond or bell (lowest) - heart - spade or leaf - club or acorn (highest). See, for example, the game of Bruus. Other expressions drawn from bridge and similar games include strong suit (any area of personal strength) and to follow suit (to imitate another's actions). See also[edit]
Notes[edit]
References[edit]
What is the probability of drawing an ace of spades or a 2 from a deck of shuffled cards co3?Probability of an ace is 1/13. Probability of a spade is 1/4. Probability of both is 1/52. What is the probability of drawing a heart and a spade?The probability of choosing a heart, P(Heart) = 13/52 = 0.25. Without replacement, you now have 51 cards left in the deck. So the probability of subsequently choosing a Spade is, P(Spade) = 13/51. What is the probability of drawing a 2 or a heart from a standard deck of cards?The chance of pulling out a heart card in now 1251 . To find the probability that both cards drawn out are hearts, multiply the two fractions together: (1352)⋅(1251)=1562652=117 . What is the probability of drawing 2 spades from a deck of 52 cards?Solution: In a playing card there are 52 cards. (i) '2' of spades: Number of favourable outcomes i.e. '2' of spades is 1 out of 52 cards. What is the probability of drawing a heart and a spade card?If you're wanting to draw a heart and a spade, then you could get the heart first, or the spade first. The probability of doing this with replacement is 2(1/4)(1/4)=1/8. Doing it without replacement is 2(1/4)(13/51)=13/102.
What is the probability of drawing a spade or a heart from a deck of 52 cards?Each deck of 52 cards has 13 hearts, 13 clubs, 13 spades and 13 diamonds. For the first draw, the probability of a heart is 13/52 = 1/4 = 0.2500. The card is not replaced, so 51 cards remain in the deck. For the second draw, the probability of a spade is 13/51 = 0.2549.
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