What is the probability of getting exactly one pair in a game of 5 card poker

In this lesson, we explain how to compute the probability of being dealt two or more cards of equal rank in stud poker. (For a brief description of stud poker, click here.)

Hands With Equal-Rank Cards

In stud poker, there are five types of hands that include two or more cards of equal rank.

• Four of a kind. Four cards of the same rank, such as 9♠, 9♥, 9♦, 9♣, 2♥
• Full house. Three cards of one rank and two of another rank, such as 9♠, 9♥, 9♦, 2♣, 2♥.
• Three of a kind. Three cards of the same rank and two cards of different ranks, such as 9♠, 9♥, 9♦, 4♣, Q♥.
• Two pair. Two cards of equal rank, two other cards of equal but different rank, and another card of different rank, such as A♠, A♥, 5♦, 5♣, 7♥.
• One pair. Two cards of equal rank and three cards of different rank, such as 8♠, 8♥, 2♦, J♣, K♥.

In this lesson, we will compute probabilities for each of these hands.

How to Compute Poker Probabilities

In a previous lesson, we explained how to compute probability for any type of poker hand. For convenience, here is a brief review:

• Count the number of possible five-card hands that can be dealt from a standard deck of 52 cards
• Count the number of ways that a particular type of poker hand can occur
• The probability of being dealt any particular type of hand is equal to the number of ways it can occur divided by the total number of possible five-card hands.

So, how do we count the number of ways that different types of poker hands can occur? We recognize that every poker hand consists of five cards, and the order in which cards are arranged does not matter. When you talk about all the possible ways to count a set of objects without regard to order, you are talking about counting combinations. Luckily, we have a formula to do that:

Counting combinations. The number of combinations of n objects taken r at a time is

nCr = n(n - 1)(n - 2) . . . (n - r + 1)/r! = n! / r!(n - r)!

In summary, we use the combination formula to count (a) the number of possible five-card hands and (b) the number of ways a particular type of hand can be dealt. To find probability, we divide the latter by the former.

Probability of Four of a Kind

Let's execute the analytical plan described above to find the probability of four of a kind.

• First, we count the number of five-card hands that can be dealt from a standard deck of 52 cards. This is a combination problem. The number of combinations is n! / r!(n - r)!. We have 52 cards in the deck so n = 52. And we want to arrange them in unordered groups of 5, so r = 5. Thus, the number of combinations is:

  52C5 = 52! / 5!(52 - 5)! = 52! / 5!47! = 2,598,960

  Hence, there are 2,598,960 distinct poker hands.
• Next, we count the number of ways that five cards can be dealt to produce four of a kind. It requires three independent choices to produce four of a kind:
  • Choose the rank of the card that appears four times in the hand. A playing card can have a rank of 2, 3, 4, 5, 6, 7, 8, 9, 10, jack, queen, king, or ace. For four of a kind, we choose 1 rank from a set of 13 ranks. The number of ways to do this is 13C1.
  • Choose one rank for the fifth card. There are 12 remaining ranks, from which we choose one. The number of ways to do this is 12C1.
  • Choose a suit for the fifth card. There are four suits, from which we choose one. The number of ways to do this is 4C1.

  The number of ways to produce one pair (Numop) is equal to the product of the number of ways to make each independent choice. Therefore,

  In poker, the probability of each type of 5-card hand can be computed by calculating the proportion of hands of that type among all possible hands.

  History[edit]

  Probability and gambling have been ideas since long before the invention of poker. The development of probability theory in the late 1400s was attributed to gambling; when playing a game with high stakes, players wanted to know what the chance of winning would be. In 1494, Fra Luca Paccioli released his work Summa de arithmetica, geometria, proportioni e proportionalita which was the first written text on probability. Motivated by Paccioli's work, Girolamo Cardano (1501-1576) made further developments in probability theory. His work from 1550, titled Liber de Ludo Aleae, discussed the concepts of probability and how they were directly related to gambling. However, his work did not receive any immediate recognition since it was not published until after his death. Blaise Pascal (1623-1662) also contributed to probability theory. His friend, Chevalier de Méré, was an avid gambler with the goal to become wealthy from it. De Méré tried a new mathematical approach to a gambling game but did not get the desired results. Determined to know why his strategy was unsuccessful, he consulted with Pascal. Pascal's work on this problem began an important correspondence between him and fellow mathematician Pierre de Fermat (1601-1665). Communicating through letters, the two continued to exchange their ideas and thoughts. These interactions led to the conception of basic probability theory. To this day, many gamblers still rely on the basic concepts of probability theory in order to make informed decisions while gambling.[1][2]

  Frequencies[edit]

  5-card poker hands[edit]

  

  The following chart enumerates the (absolute) frequency of each hand, given all combinations of 5 cards randomly drawn from a full deck of 52 without replacement. Wild cards are not considered. In this chart:

  • Distinct hands is the number of different ways to draw the hand, not counting different suits.
  • Frequency is the number of ways to draw the hand, including the same card values in different suits.
  • The Probability of drawing a given hand is calculated by dividing the number of ways of drawing the hand (Frequency) by the total number of 5-card hands (the sample space; (525)=2,598,960{\textstyle {52 \choose 5}=2,598,960}
    
    ). For example, there are 4 different ways to draw a royal flush (one for each suit), so the probability is 4/2,598,960, or one in 649,740. One would then expect to draw this hand about once in every 649,740 draws, or nearly 0.00000154% of the time.
  • Cumulative probability refers to the probability of drawing a hand as good as or better than the specified one. For example, the probability of drawing three of a kind is approximately 2.11%, while the probability of drawing a hand at least as good as three of a kind is about 2.87%. The cumulative probability is determined by adding one hand's probability with the probabilities of all hands above it.
  • The Odds are defined as the ratio of the number of ways not to draw the hand, to the number of ways to draw it. In statistics, this is called odds against. For instance, with a royal flush, there are 4 ways to draw one, and 2,598,956 ways to draw something else, so the odds against drawing a royal flush are 2,598,956 : 4, or 649,739 : 1. The formula for establishing the odds can also be stated as (1/p) - 1 : 1, where p is the aforementioned probability.
  • The values given for Probability, Cumulative probability, and Odds are rounded off for simplicity; the Distinct hands and Frequency values are exact.

  The nCr function on most scientific calculators can be used to calculate hand frequencies; entering nCr with 52 and 5, for example, yields (525)=2,598,960{\textstyle {52 \choose 5}=2,598,960} as above.

  HandDistinct handsFrequencyProbabilityCumulative probabilityOdds againstMathematical expression of absolute frequencyRoyal flush
  
  
  
  
  
  140.000154%0.000154%649,739 : 1(41){\displaystyle {4 \choose 1}}
  
  Straight flush (excluding royal flush)
  
  
  
  
  
  9360.00139%0.0015%72,192.33 : 1(101)(41)−(41){\displaystyle {10 \choose 1}{4 \choose 1}-{4 \choose 1}}
  
  Four of a kind
  
  
  
  
  
  1566240.02401%0.0256%4,165 : 1(131)(44)(121)(41){\displaystyle {13 \choose 1}{4 \choose 4}{12 \choose 1}{4 \choose 1}}
  
  Full house
  
  
  
  
  
  1563,7440.1441%0.17%693.1667 : 1(131)(43)(121)(42){\displaystyle {13 \choose 1}{4 \choose 3}{12 \choose 1}{4 \choose 2}}
  
  Flush (excluding royal flush and straight flush)
  
  
  
  
  
  1,2775,1080.1965%0.367%508.8019 : 1(135)(41)−(101)(41){\displaystyle {13 \choose 5}{4 \choose 1}-{10 \choose 1}{4 \choose 1}}
  
  Straight (excluding royal flush and straight flush)
  
  
  
  
  
  1010,2000.3925%0.76%253.8 : 1(101)(41)5−(101)(41){\displaystyle {10 \choose 1}{4 \choose 1}^{5}-{10 \choose 1}{4 \choose 1}}
  
  Three of a kind
  
  
  
  
  
  85854,9122.1128%2.87%46.32955 : 1(131)(43)(122)(41)2{\displaystyle {13 \choose 1}{4 \choose 3}{12 \choose 2}{4 \choose 1}^{2}}
  
  Two pair
  
  
  
  
  
  858123,5524.7539%7.62%20.03535 : 1(132)(42)2(111)(41){\displaystyle {13 \choose 2}{4 \choose 2}^{2}{11 \choose 1}{4 \choose 1}}
  
  One pair
  
  
  
  
  
  2,8601,098,24042.2569%49.9%1.366477 : 1(131)(42)(123)(41)3{\displaystyle {13 \choose 1}{4 \choose 2}{12 \choose 3}{4 \choose 1}^{3}}
  
  No pair / High card
  
  
  
  
  
  1,2771,302,54050.1177%100%0.9953015 : 1[(135)−(101)][(41)5−(41)]{\displaystyle \left[{13 \choose 5}-{10 \choose 1}\right]\left[{4 \choose 1}^{5}-{4 \choose 1}\right]}
  
  Total7,4622,598,960100%---0 : 1(525){\displaystyle {52 \choose 5}}
  

  The royal flush is a case of the straight flush. It can be formed 4 ways (one for each suit), giving it a probability of 0.000154% and odds of 649,739 : 1.

  When ace-low straights and ace-low straight flushes are not counted, the probabilities of each are reduced: straights and straight flushes each become 9/10 as common as they otherwise would be. The 4 missed straight flushes become flushes and the 1,020 missed straights become no pair.

  Note that since suits have no relative value in poker, two hands can be considered identical if one hand can be transformed into the other by swapping suits. For example, the hand 3♣ 7♣ 8♣ Q♠ A♠ is identical to 3♦ 7♦ 8♦ Q♥ A♥ because replacing all of the clubs in the first hand with diamonds and all of the spades with hearts produces the second hand. So eliminating identical hands that ignore relative suit values, there are only 134,459 distinct hands.

  The number of distinct poker hands is even smaller. For example, 3♣ 7♣ 8♣ Q♠ A♠ and 3♦ 7♣ 8♦ Q♥ A♥ are not identical hands when just ignoring suit assignments because one hand has three suits, while the other hand has only two—that difference could affect the relative value of each hand when there are more cards to come. However, even though the hands are not identical from that perspective, they still form equivalent poker hands because each hand is an A-Q-8-7-3 high card hand. There are 7,462 distinct poker hands.

  7-card poker hands[edit]

  In some popular variations of poker such as Texas hold 'em, a player uses the best five-card poker hand out of seven cards.

  The frequencies are calculated in a manner similar to that shown for 5-card hands, except additional complications arise due to the extra two cards in the 7-card poker hand. The total number of distinct 7-card hands is (527)=133,784,560{\textstyle {52 \choose 7}=133,784,560}

  
  . It is notable that the probability of a no-pair hand is lower than the probability of a one-pair or two-pair hand.

  The Ace-high straight flush or royal flush is slightly more frequent (4324) than the lower straight flushes (4140 each) because the remaining two cards can have any value; a King-high straight flush, for example, cannot have the Ace of its suit in the hand (as that would make it ace-high instead).

  HandFrequencyProbabilityCumulativeOdds againstMathematical expression of absolute frequencyRoyal flush
  
  
  
  
  
  4,3240.0032%0.0032%30,939 : 1(41)(472){\displaystyle {4 \choose 1}{47 \choose 2}}
  
  Straight flush (excluding royal flush)
  
  
  
  
  
  37,2600.0279%0.0311%3,589.6 : 1(91)(41)(462){\displaystyle {9 \choose 1}{4 \choose 1}{46 \choose 2}}
  
  Four of a kind
  
  
  
  
  
  224,8480.168%0.199%594 : 1(131)(483){\displaystyle {13 \choose 1}{48 \choose 3}}
  
  Full house
  
  
  
  
  
  3,473,1842.60%2.80%37.5 : 1[(132)(43)2(441)]+[(131)(122)(43)(42)2]+[(131)(121)(112)(43)(42)(41)2]{\displaystyle {\begin{aligned}&\left[{13 \choose 2}{4 \choose 3}^{2}{44 \choose 1}\right]\\+&\left[{13 \choose 1}{12 \choose 2}{4 \choose 3}{4 \choose 2}^{2}\right]\\+&\left[{13 \choose 1}{12 \choose 1}{11 \choose 2}{4 \choose 3}{4 \choose 2}{4 \choose 1}^{2}\right]\end{aligned}}}
  
  Flush (excluding royal flush and straight flush)
  
  
  
  
  
  4,047,6443.03%5.82%32.1 : 1[(41)×[(137)−217]]+[(41)×[(136)−71]×39]+[(41)×[(135)−10]×(392)]{\displaystyle {\begin{aligned}&\left[{4 \choose 1}\times \left[{13 \choose 7}-217\right]\right]\\+&\left[{4 \choose 1}\times \left[{13 \choose 6}-71\right]\times 39\right]\\+&\left[{4 \choose 1}\times \left[{13 \choose 5}-10\right]\times {39 \choose 2}\right]\end{aligned}}}
  
  Straight (excluding royal flush and straight flush)
  
  
  
  
  
  6,180,0204.62%10.4%20.6 : 1[217×[47−756−4−84]]+[71×36×990]+[10×5×4×[256−3]+10×(52)×2268]{\displaystyle {\begin{aligned}&\left[217\times \left[4^{7}-756-4-84\right]\right]\\+&{}\left[71\times 36\times 990\right]\\+&\left[10\times 5\times 4\times \left[256-3\right]+10\times {5 \choose 2}\times 2268\right]\end{aligned}}}
  
  Three of a kind
  
  
  
  
  
  6,461,6204.83%15.3%19.7 : 1[(135)−10](51)(41)[(41)4−3]{\displaystyle \left[{13 \choose 5}-10\right]{5 \choose 1}{4 \choose 1}\left[{4 \choose 1}^{4}-3\right]}
  
  Two pair
  
  
  
  
  
  31,433,40023.5%38.8%3.26 : 1[1277×10×[6×62+24×63+6×64]]+[(133)(42)3(401)]{\displaystyle {\begin{aligned}&\left[1277\times 10\times \left[6\times 62+24\times 63+6\times 64\right]\right]\\+&\left[{13 \choose 3}{4 \choose 2}^{3}{40 \choose 1}\right]\end{aligned}}}
  
  One pair
  
  
  
  
  
  58,627,80043.8%82.6%1.28 : 1[(136)−71]×6×6×990{\displaystyle \left[{13 \choose 6}-71\right]\times 6\times 6\times 990}
  
  No pair / High card
  
  
  
  
  
  23,294,46017.4%100%4.74 : 11499×[47−756−4−84]{\displaystyle 1499\times \left[4^{7}-756-4-84\right]}
  
  Total133,784,560100%---0 : 1(527){\displaystyle {52 \choose 7}}
  

  (The frequencies given are exact; the probabilities and odds are approximate.)

  Since suits have no relative value in poker, two hands can be considered identical if one hand can be transformed into the other by swapping suits. Eliminating identical hands that ignore relative suit values leaves 6,009,159 distinct 7-card hands.

  The number of distinct 5-card poker hands that are possible from 7 cards is 4,824. Perhaps surprisingly, this is fewer than the number of 5-card poker hands from 5 cards, as some 5-card hands are impossible with 7 cards (e.g. 7-high and 8-high).

  5-card lowball poker hands[edit]

  Some variants of poker, called lowball, use a low hand to determine the winning hand. In most variants of lowball, the ace is counted as the lowest card and straights and flushes don't count against a low hand, so the lowest hand is the five-high hand A-2-3-4-5, also called a wheel. The probability is calculated based on (525)=2,598,960{\textstyle {52 \choose 5}=2,598,960}, the total number of 5-card combinations. (The frequencies given are exact; the probabilities and odds are approximate.)

  HandDistinct handsFrequencyProbabilityCumulativeOdds against5-high11,0240.0394%0.0394%2,537.05 : 16-high55,1200.197%0.236%506.61 : 17-high1515,3600.591%0.827%168.20 : 18-high3535,8401.38%2.21%71.52 : 19-high7071,6802.76%4.96%35.26 : 110-high126129,0244.96%9.93%19.14 : 1Jack-high210215,0408.27%18.2%11.09 : 1Queen-high330337,92013.0%31.2%6.69 : 1King-high495506,88019.5%50.7%4.13 : 1Total1,2871,317,88850.7%50.7%0.97 : 1

  As can be seen from the table, just over half the time a player gets a hand that has no pairs, threes- or fours-of-a-kind. (50.7%)

  If aces are not low, simply rotate the hand descriptions so that 6-high replaces 5-high for the best hand and ace-high replaces king-high as the worst hand.

  Some players do not ignore straights and flushes when computing the low hand in lowball. In this case, the lowest hand is A-2-3-4-6 with at least two suits. Probabilities are adjusted in the above table such that "5-high" is not listed, "6-high" has 781,824 distinct hands, and "King-high" has 21,457,920 distinct hands, respectively. The Total line also needs adjusting.

  How many different 5 card hands have exactly one pair?

  5-card poker hands.

  What is the probability of getting a straight in 5 card poker?

  (52−5)! (5!) =2.6 million possible hands. 0.4% chance of pulling a straight on 5 cards.

  What is the probability of getting a pair in poker?

  The probability of a pair in poker is ~42%. The chances of making a full house poker probability is less than 1% (~0.1441%) The probability in poker Texas Hold'em of making a royal flush is just 1 in 649,740 hands! The likelihood of a straight flush in poker is 1 in 72,193 hands or 0.00139%.

  What is the probability that a five

  The probability that a 5 card poker hand contains exactly one ace is ≈0.2995.