Probability and Poker

In the standard game of poker, each player gets 5 cards and places a bet, hoping his cards are better than the other players' hands.

The game is played with a pack containing 52 cards in 4 suits, consisting of:

13 hearts:
13 diamonds
13 clubs:

♥ 2 3 4 5 6 7 8 9 10 J Q K A
♦ 2 3 4 5 6 7 8 9 10 J Q K A
♣ 2 3 4 5 6 7 8 9 10 J Q K A
♠ 2 3 4 5 6 7 8 9 10 J Q K A

The number of different possible poker hands is found by counting the number of ways that 5 cards can be selected from 52 cards, where the order is not important. It is a combination, so we use `C r^n`.

The number of possible poker hands

Royal Flush

The best hand (because of the low probability that it will occur) is the royal flush, which consists of 10, J, Q, K, A of the same suit. There are only 4 ways of getting such a hand (because there

are 4 suits), so the probability of being dealt a royal flush is

Straight Flush

The next most valuable type of hand is a straight flush, which is 5 cards in order, all of the same suit.

For example, 2♣, 3♣, 4♣, 5♣, 6♣ is a straight flush.

For each suit there are 10 such straights (the one starting with Ace, the one starting with 2, the one starting with 3, . through to the one starting at 10) and there are 4 suits, so there are 40 possible straight flushes.

The probability of being dealt a straight flush is

[Note: There is some overlap here since the straight flush starting at 10 is the same as the royal flush. So strictly there are 36 straight flushes (4 × 9) if we don't count the royal flush. The probability of getting a straight flush then is 36/2,598,960 = 0.00001385 .]

The table below lists the number of possible ways that different types of hands can arise and their probability of occurrence.

5 Card Poker probabilities

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.

Frequency of 5-card poker hands

The following 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. The probability of drawing a given hand is calculated by dividing the number of ways of drawing the hand by the total number of 5-card hands (the sample space, five-card hands). The odds are defined as the ratio (1/p) - 1 : 1, where p is the probability. Note that the cumulative column contains the probability of being dealt that hand or any of the hands ranked higher than it. (The frequencies given are exact; the probabilities and odds are approximate.)

The nCr function on most scientific calculators can be used to calculate hand frequencies; entering ​nCr​ with ​52​ and ​5​, for example, yields as above.

Hand Frequency Approx. Probability Approx. Cumulative Approx. Odds Mathematical expression of absolute frequency
Royal flush 4 0.000154% 0.000154% 649,739 : 1
Straight flush (excluding royal flush) 36 0.00139% 0.00154% 72,192.33 : 1
Four of a kind 624 0.0240% 0.0256% 4,164 : 1
Full house 3,744 0.144% 0.170% 693.2 : 1
Flush (excluding royal flush and straight flush) 5,108 0.197% 0.367% 507.8 : 1
Straight (excluding royal flush and straight flush) 10,200 0.392% 0.76% 253.8 : 1
Three of a kind 54,912 2.11% 2.87% 46.3 : 1
Two pair 123,552 4.75% 7.62% 20.03 : 1
One pair 1,098,240 42.3% 49.9% 1.36 : 1
No pair / High card 1,302,540 50.1% 100% .995 : 1
Total 2,598,960 100% 100% 1 : 1

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 twothat 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.

Derivation of frequencies of 5-card poker hands

of the binomial coefficients and their interpretation as the number of ways of choosing elements from a given set. See also: sample space and event (probability theory).

• Straight flush Each straight flush is uniquely determined by its highest ranking card; and these ranks go from 5 (A-2-3-4-5) up to A (10-J-Q-K-A) in each of the 4 suits. Thus, the total number of straight flushes is:
• Royal straight flush A royal straight flush is a subset of all straight flushes in which the ace is the highest card (ie 10-J-Q-K-A in any of the four suits). Thus, the total number of royal straight flushes is or simply . Note: this means that the total number of non-Royal straight flushes is 36.
• Four of a kind Any one of the thirteen ranks can form the four of a kind by selecting all four of the suits in that rank. The final card can have any one of the twelve remaining ranks, and any suit. Thus, the total number of four-of-a-kinds is:
• Full house The full house comprises a triple (three of a kind) and a pair. The triple can be any one of the thirteen ranks, and consists of three of the four suits. The pair can be any one of the remaining twelve ranks, and consists of two of the four suits. Thus, the total number of full houses is:
• Flush The flush contains any five of the thirteen ranks, all of which belong to one of the four suits, minus the 40 straight flushes. Thus, the total number of flushes is:
• Straight The straight consists of any one of the ten possible sequences of five consecutive cards, from 5-4-3-2-A to A-K-Q-J-10. Each of these five cards can have any one of the four suits. Finally, as with the flush, the 40 straight flushes must be excluded, giving:
• Three of a kind Any of the thirteen ranks can form the three of a kind, which can contain any three of the four suits. The remaining two cards can have any two of the remaining twelve ranks, and each can have any of the four suits. Thus, the total number of three-of-a-kinds is:
• Two pair The pairs can have any two of the thirteen ranks, and each pair can have two of the four suits. The final card can have any one of the eleven remaining ranks, and any suit. Thus, the total number of two-pairs is:
• Pair The pair can have any one of the thirteen ranks, and any two of the four suits. The remaining three cards can have any three of the remaining twelve ranks, and each can have any of the four suits. Thus, the total number of pair hands is:
• No pair A no-pair hand contains five of the thirteen ranks, discounting the ten possible straights, and each card can have any of the four suits, discounting the four possible flushes. Alternatively, a no-pair hand is any hand that does not fall into one of the above categories; that is, any way to choose five out of 52 cards, discounting all of the above hands. Thus, the total number of no-pair hands is:
• Any five card poker hand The total number of five card hands that can be drawn from a deck of cards is found using a combination selecting five cards, in any order where n refers to the number of items that can be selected and r to the sample size; the ! is the factorial operator:

The Probability of Being Dealt a Royal Flush in Poker

If you watch any movie that involves poker, it seems like it s only a matter of time before a royal flush makes an appearance. This is a poker hand that has a very specific composition: the ten, jack, queen, king and ace, all of the same suit. Typically the hero of the movie is dealt this hand and it is revealed in a dramatic fashion. A royal flush is the highest ranked hand in the card game of poker. Due to the specifications for this hand, it is very difficult to be dealt a royal flush.

Basic Assumptions and Probability

There is a multitude of different ways that poker can be played. For our purposes, we will assume that a player is dealt five cards from a standard 52 card deck. No cards are wild, and the player keeps all of the cards that are dealt to him or her.

To calculate the probability of being dealt a royal flush, we need to know two numbers:

• The total number of possible poker hands
• The total number of ways that a royal flush can be dealt.

Once we know these two numbers, the probability of being dealt a royal flush is a simple calculation. All that we have to do is to divide the second number by the first number.

Number of Poker Hands

Some of the techniques of combinatorics, or the study of counting, can be applied to calculate the total number of poker hands. It is important to note that the order in which the cards are dealt to us does not matter. Since the order does not matter, this means that each hand is a combination of five cards from a total of 52. We use the formula for combinations and see that there are a total number of C( 52, 5 ) = 2,598,960 possible distinct hands.

Royal Flush

A royal flush is a flush. This means that all of the cards must be of the same suit. There are a number of different kinds of flushes. Unlike most flushes, in a royal flush, the value of all five cards are completely specified. The cards in one's hand must be a ten, jack, queen, king and ace all of the same suit.

For any given suit there is only one combination of cards with these cards. Since there are four suits of hearts, diamonds, clubs, and spades, there are only four possible royal flushes that can be dealt.

The Probability of a Royal Flush

We can already tell from the numbers above that a royal flush is unlikely to be dealt. Of the nearly 2.6 million poker hands, only four of them are royal flushes. These nearly 2.6 hands are uniformly distributed. Due to the shuffling of the cards, every one of these hands is equally likely to be dealt to a player.

The probability of being dealt a royal flush is the number of royal flushes divided by the total number of poker hands. We now carry out the division and see that a royal flush is rare indeed. There is only a probability of 4/2,598,960 = 1/649,740 = 0.00015% of being dealt this hand.

Much like very large numbers, a probability that is this small is hard to wrap your head around. One way to put this number in perspective is to ask how long it would take to go through 649,740 poker hands. If you were dealt 20 hands of poker every night of the year, then this would only amount to 7300 hands per year. in 89 years you should only expect to see one royal flush. So this hand is not as common as what the movies might make us believe.

How to Compute the Probability of Equal-Rank Cards in Stud Poker

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

How to Compute Poker Probabilities

• 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

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

• 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 four of a kind (Num4) is equal to the product of the number of ways to make each independent choice. Therefore,

The probability of being dealt four of a kind is 0.0002400960384. On average, four of a kind is dealt one time in every 4,165 deals.

Probability of a Full House

• First, count the number of five-card hands that can be dealt from a standard deck of 52 cards. We did this in the previous section, and found that there are 2,598,960 distinct poker hands.
• Next, count the number of ways that five cards can be dealt to produce a full house. It requires four independent choices to produce a full house:
• Choose the rank of cards 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 a full house, we choose 2 ranks from a set of 13 ranks. The number of ways to do this is 13C2.
• Choose one rank for the three-card combination. There are 2 ranks in a full house, from which we choose one. The number of ways to do this is 2C1.
• Choose suits for the three-card combination. There are four suits, from which we choose three. The number of ways to do this is 4C3.
• Choose suits for the two-card combination. There are four suits, from which we choose two. The number of ways to do this is 4C2.

The number of ways to produce full house (Numfh) is equal to the product of the number of ways to make each independent choice. Therefore,

Based on these results, we can project that a full house will be dealt, on average, approximately one time in every 694 deals.

Probability of Three of a Kind

• First, count the number of five-card hands that can be dealt from a standard deck of 52 cards. We did this previously, and found that there are 2,598,960 distinct poker hands.
• Next, count the number of ways that five cards can be dealt to produce three of a kind. It requires five independent choices to produce three of a kind:
• Choose the rank for cards of matching rank. A playing card can have a rank of 2, 3, 4, 5, 6, 7, 8, 9, 10, jack, queen, king, or ace. For three of a kind, we choose 1 rank from a set of 13 ranks. The number of ways to do this is 13C1.
• Choose the rank of non-matching cards. There are 12 remaining ranks, from which we choose two. The number of ways to do this is 12C2.
• Choose suits for the three-card combination. There are four suits, from which we choose three. The number of ways to do this is 4C3.
• Choose a suit for one of the non-matching cards. There are four suits, from which we choose one. The number of ways to do this is 4C1.
• Choose a suit for the other non-matching 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 three of a kind (Num3) is equal to the product of the number of ways to make each independent choice. Therefore,

In stud poker, players get three of a kind about one time in every 47 deals.

Probability of Two Pair

• First, count the number of five-card hands that can be dealt from a standard deck of 52 cards. We did this previously, and found that there are 2,598,960 distinct poker hands.
• Next, count the number of ways that five cards can be dealt to produce two pair. It requires five independent choices to produce two pair:
• Choose the rank for cards of matching rank. A playing card can have a rank of 2, 3, 4, 5, 6, 7, 8, 9, 10, jack, queen, king, or ace. For two pair, we choose 2 ranks from a set of 13 ranks. The number of ways to do this is 13C2.
• Choose the rank of the remaining non-matching card. There are 11 remaining ranks, from which we choose one. The number of ways to do this is 11C1.
• Choose suits for the first two-card combination. There are four suits, from which we choose two. The number of ways to do this is 4C2.
• Choose suits for the second two-card combination. There are four suits, from which we choose two. The number of ways to do this is 4C2.
• Choose a suit for the non-matching 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 two pair (Numtp) is equal to the product of the number of ways to make each independent choice. Therefore,

On average, players get two pair about one time in every 21 deals.

Probability of One Pair

• First, count the number of five-card hands that can be dealt from a standard deck of 52 cards. We did this previously, and found that there are 2,598,960 distinct poker hands.
• Next, count the number of ways that five cards can be dealt to produce one pair. It requires six independent choices to produce one pair:
• Choose the rank for cards of matching rank. A playing card can have a rank of 2, 3, 4, 5, 6, 7, 8, 9, 10, jack, queen, king, or ace. For one pair, we choose 1 rank from a set of 13 ranks. The number of ways to do this is 13C1.
• Choose a rank for each of the remaining non-matching cards. There are 12 remaining ranks and three non-matching cards, so we choose three ranks from the remaining 12. The number of ways to do this is 12C3.
• Choose suits for the cards of matching rank. There are four suits, from which we choose two. The number of ways to do this is 4C2.
• Choose a suit for the first non-matching rank. There are four suits, from which we choose one. The number of ways to do this is 4C1.
• Choose a suit for the second non-matching rank. There are four suits, from which we choose one. The number of ways to do this is 4C1.
• Choose a suit for the third non-matching rank. 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 stud poker, on any given hand, there is about a 42% chance that a player will be dealt one pair.

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