What is the probability of drawing 4 cards from a standard 52 card deck and getting 4 of a kind

Important Notes

• The sample space for a set of cards is 52 as there are 52 cards in a deck. This makes the denominator for finding the probability of drawing a card as 52.
• Learn more about related terminology of probability to solve problems on card probability better. 

The suits which are represented by red cards are hearts and diamonds while the suits represented by black cards are spades and clubs.

There are 26 red cards and 26 black cards. 

Let's learn about the suits in a deck of cards.

Suits in a deck of cards are the representations of red and black color on the cards.

Based on suits, the types of cards in a deck are: 

There are 52 cards in a deck.

Each card can be categorized into 4 suits constituting 13 cards each.

These cards are also known as court cards.

They are Kings, Queens, and Jacks in all 4 suits.

All the cards from 2 to 10 in any suit are called the number cards. 

These cards have numbers on them along with each suit being equal to the number on number cards. 

There are 4 Aces in every deck, 1 of every suit. 

Tips and Tricks

• There are 13 cards of each suit, consisting of 1 Ace, 3 face cards, and 9 number cards.
• There are 4 Aces, 12 face cards, and 36 number cards in a 52 card deck.
• Probability of drawing any card will always lie between 0 and 1.
• The number of spades, hearts, diamonds, and clubs is same in every pack of 52 cards.

Now that you know all about facts about a deck of cards, you can draw a card from a deck and find its probability easily.

How to Determine the Probability of Drawing a Card?

Let's learn how to find probability first.

Now you know that probability is the ratio of number of favorable outcomes to the number of total outcomes, let's apply it here.

Examples

Example 1: What is the probability of drawing a king from a deck of cards?

Solution: Here the event E is drawing a king from a deck of cards.

There are 52 cards in a deck of cards. 

Hence, total number of outcomes = 52

The number of favorable outcomes = 4 (as there are 4 kings in a deck)

Hence, the probability of this event occuring is 

P(E) = 4/52 = 1/13

Example 2: What is the probability of drawing a black card from a pack of cards?

Solution: Here the event E is drawing a black card from a pack of cards.

The total number of outcomes = 52

The number of favorable outcomes = 26

Hence, the probability of event occuring is 

P(E) = 26/52 = 1/2

Solved Examples

Jessica has drawn a card from a well-shuffled deck. Help her find the probability of the card either being red or a King.

Solution

Jessica knows here that event E is the card drawn being either red or a King.

The total number of outcomes = 52

There are 26 red cards, and 4 cards which are Kings.

However, 2 of the red cards are Kings.

If we add 26 and 4, we will be counting these two cards twice.

Thus, the correct number of outcomes which are favorable to E is

26 + 4 - 2 = 28

Hence, the probability of event occuring is

P(E) = 28/52 = 7/13

Help Diane determine the probability of the following:

• Drawing a Red Queen
• Drawing a King of Spades
• Drawing a Red Number Card 

Solution

Diane knows here the events E1, E2, and E3 are Drawing a Red Queen, Drawing a King of Spades, and Drawing a Red Number Card.

The total number of outcomes in every case = 52

There are 26 red cards, of which 2 are Queens.

Hence, the probability of event E1 occuring is

P(E1) = 2/52 = 1/26

There are 13 cards in each suit, of which 1 is King.

Hence, the probability of event E2 occuring is

P(E2) = 1/52 

• Drawing a Red Number Card

There are 9 number cards in each suit and there are 2 suits which are red in color. 

There are 18 red number cards.

Hence, the probability of event E3 occuring is

P(E3) = 18/52 = 9/26 

Interactive Questions

Here are a few activities for you to practice.

Select/Type your answer and click the "Check Answer" button to see the result.

We hope you enjoyed learning about probability of drawing a card from a pack of 52 cards with the practice questions. Now you will easily be able to solve problems on number of cards in a deck, face cards in a deck, 52 card deck, spades hearts diamonds clubs in pack of cards. Now you can draw a card from a deck and find its probability easily .

The mini-lesson targeted the fascinating concept of card probability. The math journey around card probability starts with what a student already knows, and goes on to creatively crafting a fresh concept in the young minds. Done in a way that is not only relatable and easy to grasp, but will also stay with them forever. Here lies the magic with Cuemath.

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At Cuemath, our team of math experts is dedicated to making learning fun for our favorite readers, the students! Through an interactive and engaging learning-teaching-learning approach, the teachers explore all angles of a topic. Be it problems, online classes, videos, or any other form of relation, it’s the logical thinking and smart learning approach that we, at Cuemath, believe in.

We find the ratio of the favorable outcomes as per the condition of drawing the card to the total number of outcomes, i.e, 52.

2. What is the probability of drawing any face card?

Probability of drawing any face card is 6/26.

3. What is the probability of drawing a red card?

Probability of drawing a red card is 1/2.

4. What is the probability of drawing a king or a red card?

Probability of drawing a king or a red card is 7/13.

5. What is the probability of drawing a king or a queen?

The probability of drawing a king or a queen is 2/13.

6. What are the 5 rules of probability?

The 5 rules of probability are:

For any event E, the probability of occurence of E will always lie between 0 and 1

The sum of probabilities of every possible outcome will always be 1

The sum of probability of occurence of E and probability of E not occuring will always be 1

When any two events are not disjoint, the probability of occurence of A and B is not 0 while when two events are disjoint, the probability of occurence of A and B is 0.

As per this rule, P(A or B) = (P(A) + P(B) - P(A and B)).

7. What is the probability of drawing a king of hearts?

Probability of drawing a king of hearts is 1/52.

8. Is Ace a face card in probability?

No, Ace is not a face card in probability.

9. What is the probability it is not a face card?

The probability it is not a face card is 10/13.

10. How many black non-face cards are there in a deck?

There are 20 black non-face cards in a deck.

Chances of card combinations in poker

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

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

5-card poker hands

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; ( 52 5 ) = 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 0.000154% 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 ( 52 5 ) = 2 , 598 , 960 {\textstyle {52 \choose 5}=2,598,960} as above.

Hand	Distinct hands	Frequency	Probability	Cumulative probability	Odds against	Mathematical expression of absolute frequency
Royal flush	1	4	0.000154%	0.000154%	649,739 : 1	( 4 1 ) {\displaystyle {4 \choose 1}}
Straight flush (excluding royal flush)	9	36	0.00139%	0.0015%	72,192.33 : 1	( 10 1 ) ( 4 1 ) − ( 4 1 ) {\displaystyle {10 \choose 1}{4 \choose 1}-{4 \choose 1}}
Four of a kind	156	624	0.02401%	0.0256%	4,165 : 1	( 13 1 ) ( 4 4 ) ( 12 1 ) ( 4 1 ) {\displaystyle {13 \choose 1}{4 \choose 4}{12 \choose 1}{4 \choose 1}}
Full house	156	3,744	0.1441%	0.17%	693.1667 : 1	( 13 1 ) ( 4 3 ) ( 12 1 ) ( 4 2 ) {\displaystyle {13 \choose 1}{4 \choose 3}{12 \choose 1}{4 \choose 2}}
Flush (excluding royal flush and straight flush)	1,277	5,108	0.1965%	0.367%	508.8019 : 1	( 13 5 ) ( 4 1 ) − ( 10 1 ) ( 4 1 ) {\displaystyle {13 \choose 5}{4 \choose 1}-{10 \choose 1}{4 \choose 1}}
Straight (excluding royal flush and straight flush)	10	10,200	0.3925%	0.76%	253.8 : 1	( 10 1 ) ( 4 1 ) 5 − ( 10 1 ) ( 4 1 ) {\displaystyle {10 \choose 1}{4 \choose 1}^{5}-{10 \choose 1}{4 \choose 1}}
Three of a kind	858	54,912	2.1128%	2.87%	46.32955 : 1	( 13 1 ) ( 4 3 ) ( 12 2 ) ( 4 1 ) 2 {\displaystyle {13 \choose 1}{4 \choose 3}{12 \choose 2}{4 \choose 1}^{2}}
Two pair	858	123,552	4.7539%	7.62%	20.03535 : 1	( 13 2 ) ( 4 2 ) 2 ( 11 1 ) ( 4 1 ) {\displaystyle {13 \choose 2}{4 \choose 2}^{2}{11 \choose 1}{4 \choose 1}}
One pair	2,860	1,098,240	42.2569%	49.9%	1.366477 : 1	( 13 1 ) ( 4 2 ) ( 12 3 ) ( 4 1 ) 3 {\displaystyle {13 \choose 1}{4 \choose 2}{12 \choose 3}{4 \choose 1}^{3}}
No pair / High card	1,277	1,302,540	50.1177%	100%	0.9953015 : 1	[ ( 13 5 ) − ( 10 1 ) ] [ ( 4 1 ) 5 − ( 4 1 ) ] {\displaystyle \left[{13 \choose 5}-{10 \choose 1}\right]\left[{4 \choose 1}^{5}-{4 \choose 1}\right]}
Total	7,462	2,598,960	100%	---	0 : 1	( 52 5 ) {\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

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 ( 52 7 ) = 133 , 784 , 560 {\textstyle {52 \choose 7}=133,784,560}

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).

Hand	Frequency	Probability	Cumulative	Odds against	Mathematical expression of absolute frequency
Royal flush	4,324	0.0032%	0.0032%	30,939 : 1	( 4 1 ) ( 47 2 ) {\displaystyle {4 \choose 1}{47 \choose 2}}
Straight flush (excluding royal flush)	37,260	0.0279%	0.0311%	3,589.6 : 1	( 9 1 ) ( 4 1 ) ( 46 2 ) {\displaystyle {9 \choose 1}{4 \choose 1}{46 \choose 2}}
Four of a kind	224,848	0.168%	0.199%	594 : 1	( 13 1 ) ( 48 3 ) {\displaystyle {13 \choose 1}{48 \choose 3}}
Full house	3,473,184	2.60%	2.80%	37.5 : 1	[ ( 13 2 ) ( 4 3 ) 2 ( 44 1 ) ] + [ ( 13 1 ) ( 12 2 ) ( 4 3 ) ( 4 2 ) 2 ] + [ ( 13 1 ) ( 12 1 ) ( 11 2 ) ( 4 3 ) ( 4 2 ) ( 4 1 ) 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)	4,047,644	3.03%	5.82%	32.1 : 1	[ ( 4 1 ) × [ ( 13 7 ) − 217 ] ] + [ ( 4 1 ) × [ ( 13 6 ) − 71 ] × 39 ] + [ ( 4 1 ) × [ ( 13 5 ) − 10 ] × ( 39 2 ) ] {\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)	6,180,020	4.62%	10.4%	20.6 : 1	[ 217 × [ 4 7 − 756 − 4 − 84 ] ] + [ 71 × 36 × 990 ] + [ 10 × 5 × 4 × [ 256 − 3 ] + 10 × ( 5 2 ) × 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	6,461,620	4.83%	15.3%	19.7 : 1	[ ( 13 5 ) − 10 ] ( 5 1 ) ( 4 1 ) [ ( 4 1 ) 4 − 3 ] {\displaystyle \left[{13 \choose 5}-10\right]{5 \choose 1}{4 \choose 1}\left[{4 \choose 1}^{4}-3\right]}
Two pair	31,433,400	23.5%	38.8%	3.26 : 1	[ 1277 × 10 × [ 6 × 62 + 24 × 63 + 6 × 64 ] ] + [ ( 13 3 ) ( 4 2 ) 3 ( 40 1 ) ] {\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	58,627,800	43.8%	82.6%	1.28 : 1	[ ( 13 6 ) − 71 ] × 6 × 6 × 990 {\displaystyle \left[{13 \choose 6}-71\right]\times 6\times 6\times 990}
No pair / High card	23,294,460	17.4%	100%	4.74 : 1	1499 × [ 4 7 − 756 − 4 − 84 ] {\displaystyle 1499\times \left[4^{7}-756-4-84\right]}
Total	133,784,560	100%	---	0 : 1	( 52 7 ) {\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

See also: Lowball (poker)

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 ( 52 5 ) = 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.)

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 one distinct hand, and "King-high" having 330 distinct hands, respectively. The Total line also needs adjusting.

7-card lowball poker hands

See also: Lowball (poker)

In some variants of poker a player uses the best five-card low hand selected from seven cards. 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 ( 52 7 ) = 133 , 784 , 560 {\textstyle {52 \choose 7}=133,784,560} , the total number of 7-card combinations.

The table does not extend to include five-card hands with at least one pair. Its "Total" represents the 95.4% of the time that a player can select a 5-card low hand without any pair.

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

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.

See also

• 
  Games portal

• Probability
• Odds
• Sample space
• Event (probability theory)
• Binomial coefficient
• Combination
• Permutation
• Combinatorial game theory
• Game complexity
• Set theory
• Gaming mathematics

References

1. ^ "Probability Theory". Science Clarified. Retrieved 7 December 2015.
2. ^ "Brief History of Probability". teacher link. Retrieved 7 December 2015.

External links

• Brian Alspach's mathematics and poker page[dead link]
• MathWorld: Poker
• Poker probabilities including conditional calculations
• Numerous poker probability tables
• 5, 6, and 7 card poker probabilities

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