3 Card Poker Hands

• 3 Card Poker Hands Ranking
• 3 Card Poker Hands Order

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

Three cards in sequence of the same suit. An Ace can count as either high or low - e.g. A-K-Q or 3-2-A Ties are broken by the highest ranked card in the hand except 3 is used for a 3, 2, Ace straight flush. 3 Card Poker Strategy Optimal strategy for 3 Card Poker indicates that you should generally play all hands of a value of Q-6-4 or better and fold everything else. This casino table game will always give the house a slight edge of around 2% over the long-term.

A free online version of 3 Card Poker with 6-card Bonus. Test out your 3 Card Poker strategy here. Practice for Vegas. Play Three Card Poker for Free. The winner is determined according to the order of 3 card poker hand ranks. If the player’s hand wins, he gets paid 1:1 on both the ante bet and the play bet. If the dealer’s hand wins, both bets are lost. It is possible for the hands to tie.

Working with poker hands is an excellent way to illustrate the counting techniques covered previously in this blog – multiplication principle, permutation and combination (also covered here). There are 2,598,960 many possible 5-card Poker hands. Thus the probability of obtaining any one specific hand is 1 in 2,598,960 (roughly 1 in 2.6 million). The probability of obtaining a given type of hands (e.g. three of a kind) is the number of possible hands for that type over 2,598,960. Thus this is primarily a counting exercise.

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Preliminary Calculation

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.

These are the same hand. Order is not important.

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.

The notation is called the binomial coefficient and is pronounced “n choose r”, which is identical to the number of -element subsets of a set with objects. Other notations for are , and . 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 2,598,960. 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 0.05% chance of happening). The numerator 1,287 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 nothing), is 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? The probability of getting all 5 cards in another suit (say heart) would also be 1287/2598960. 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 0.2% chance of happening). 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? The answer is 22308/2598960 = 0.008583433. The number of “3 diamond, 2 heart” hands is calculated as follows:

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 any case, the number of possible ways to carry out that 2-step (or 3-step) process is to multiply all the possibilities together.

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The Poker Hands

Here’s a ranking chart of the Poker hands.

The chart lists the rankings with an example for each ranking. The examples are a good reminder of the definitions. The highest ranking of them all is the royal flush, which consists of 5 consecutive cards in one suit with the highest card being Ace. There is only one such hand in each suit. Thus the chance for getting a royal flush is 4 in 2,598,960.

3 Card Poker Hands Ranking

Royal flush is a specific example of a straight flush, which consists of 5 consecutive cards in one suit. There are 10 such hands in one suit. So there are 40 hands for straight flush in total. A flush is a hand with 5 cards in the same suit but not in consecutive order (or not in sequence). Thus the requirement for flush is considerably more relaxed than a straight flush. A straight is like a straight flush in that the 5 cards are in sequence but the 5 cards in a straight are not of the same suit. For a more in depth discussion on Poker hands, see the Wikipedia entry on Poker hands.

The counting for some of these hands is done in the next section. The definition of the hands can be inferred from the above chart. For the sake of completeness, the following table lists out the definition.

Definitions of Poker Hands

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3 Card Poker Hands Order

Counting Poker Hands

Straight Flush
Counting from A-K-Q-J-10, K-Q-J-10-9, Q-J-10-9-8, …, 6-5-4-3-2 to 5-4-3-2-A, there are 10 hands that are in sequence in a given suit. So there are 40 straight flush hands all together.

Four of a Kind
There is only one way to have a four of a kind for a given rank. The fifth card can be any one of the remaining 48 cards. Thus there are 48 possibilities of a four of a kind in one rank. Thus there are 13 x 48 = 624 many four of a kind in total.

Full House
Let’s fix two ranks, say 2 and 8. How many ways can we have three of 2 and two of 8? We are choosing 3 cards out of the four 2’s and choosing 2 cards out of the four 8’s. That would be = 4 x 6 = 24. But the two ranks can be other ranks too. How many ways can we pick two ranks out of 13? That would be 13 x 12 = 156. So the total number of possibilities for Full House is

Note that the multiplication principle is at work here. When we pick two ranks, the number of ways is 13 x 12 = 156. Why did we not use = 78?

Flush
There are = 1,287 possible hands with all cards in the same suit. Recall that there are only 10 straight flush on a given suit. Thus of all the 5-card hands with all cards in a given suit, there are 1,287-10 = 1,277 hands that are not straight flush. Thus the total number of flush hands is 4 x 1277 = 5,108.

Straight
There are 10 five-consecutive sequences in 13 cards (as shown in the explanation for straight flush in this section). In each such sequence, there are 4 choices for each card (one for each suit). Thus the number of 5-card hands with 5 cards in sequence is . Then we need to subtract the number of straight flushes (40) from this number. Thus the number of straight is 10240 – 10 = 10,200.

Three of a Kind
There are 13 ranks (from A, K, …, to 2). We choose one of them to have 3 cards in that rank and two other ranks to have one card in each of those ranks. The following derivation reflects all the choosing in this process.

Two Pair and One Pair
These two are left as exercises.

High Card
The count is the complement that makes up 2,598,960.

The following table gives the counts of all the poker hands. The probability is the fraction of the 2,598,960 hands that meet the requirement of the type of hands in question. Note that royal flush is not listed. This is because it is included in the count for straight flush. Royal flush is omitted so that he counts add up to 2,598,960.

Probabilities of Poker Hands

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2017 – Dan Ma

This table game is one of the simplest types of casino poker. Unlike many other types of poker, both the dealer and the player are dealt only three cards. In BetVoyager’s version of this poker game, we have altered the standard rules in order to increase payouts when the player wins (see table 1). Furthermore, the player can play on four boxes at once and view all his cards as soon as they are dealt. The basic rules of casino poker are described in our Guide section.

BetVoyager online casino features our conventional version of 3 Card Poker and a version of this casino game with no house edge.

Main Features of 3 Card Poker

1. The player can play on up to four boxes in three card poker. The player can place an ANTE bet and/or a bet on the bonus (called Pair Plus) on any box.

2. Each box in play is dealt three cards. The cards on all boxes are turned face-up immediately after the deal in 3 Card Poker. The dealer also receives three cards, which are dealt face-down.

3. Poker hands are not ranked in the same order as 5-card poker. 3-card poker hands are ranked in the following order of seniority:

   • Straight Flush
   • Three of a Kind
   • Straight
   • Flush
   • Pair
   • Ace High or Less

   The hand A-K-Q is the highest straight and the hand 3-2-A is the lowest one in three card poker.

4. If the player made his ANTE and has a straight or higher, he receives an ANTE Bonus (table 1). This payout is made regardless of the strength of the dealer’s hand.

5. On each box, the player can either fold and lose his ANTE, or continue the game by making a BET equal to the ANTE.

6. The dealer needs a queen high or better to qualify. If the dealer doesn’t qualify, the ANTE pays out 1:1 while the BET doesn’t pay anything. If the dealer has a qualifying hand but the player’s hand is better, both bets pay out 1:1.

7. If you play more than one box, the ante amounts cannot be 5 times more or less than each other.

Basic Rules of 3 Card Poker

1. Before the cards are dealt, the player chooses which boxes he wants to play on and makes his ANTE. In addition to this basic bet on the box (or instead of it), the player can also make a Pair Plus bet. With this bet, the player is betting on receiving a pair or higher in his original hand.

2. A sign on the table indicates the range of bets permitted (minimum and maximum). The size of a Pair Plus bet is independent of the size of the ANTE, but both of these bets must fall within the table’s range.

3. Each box has the following three figures: a square, an ellipse, and a rectangle. The ANTE is placed on the ellipse and the Pair Plus bet is placed on the square. The rectangle is intended for the BET (see paragraph 7).

4. To make a bet in three card poker, it is first necessary to choose a chip with the desired face-value. At the beginning of the game, this chip is chosen by default, and the chip with the minimum face-value fixes itself onto the cursor. If the player wants a chip with a different face-value, he can select it by clicking on that chip with the cursor. The player can also use the scroll wheel on his mouse to change the chip’s value. After the chip is selected, the player can place a bet by clicking the mouse on the desired part of the game table and his bet will appear there. Each subsequent click adds another chip of the selected face-value to the bet. If the player wants to remove the last chip from his bet, he can remove the chip fixed onto the cursor by clicking the cursor on any empty part of the game window and then clicking on this bet.

5. The 'Clear' button can be pressed to remove all the bets from the table. If the player wishes to repeat his bets from the previous hand, he can press the 'Rebet' button. The 'Deal' button will begin the game after all bets have been made.

6. After the cards have been dealt, successive play on boxes begins. First, the Pair Plus bet is settled. The payouts for this bet are shown in table 1. If the player made his ANTE and has one of the three best hands — a straight flush, three-of-a-kind or a straight — he immediately receives an ANTE Bonus.

7. The player has two options on each box:

   • raise the initial wager by making a BET equal to the ANTE (the Bet button)
   • fold and lose the ANTE (the Fold button)

8. When the player has made a decision on his boxes, the dealer’s cards are turned over and the final results are determined. The dealer qualifies with a queen high or better, and his minimal qualifying hand is Q-3-2 in 3 Card poker. If the dealer doesn’t qualify, the ANTE pays out 1:1 and the BETS are not paid.

9. If the dealer qualifies and his combination is better than the player’s, the player loses his ANTE and his BET. If the player and the dealer have hands of equal value, the game ends in a draw and the ANTE and BET are returned to the player. If the player has the better hand, both his bets pay out 1:1.

Table 1. Bonus payout for the ANTE and the Pair Plus in 3 Card Poker

3 Card Poker with No House Edge features a slightly different payout scheme. It can be viewed in the game window of this poker game. The casino doesn’t have an advantage on either the ante wager or the Pair Plus bet.