3 Card Poker Hands
Besides, 3 card poker games use a 52 card deck, and three cards are offered to the player and the dealer. The three-card poker game’s main objective is to make a three-card poker hand better than the dealer’s hand. There are also two optional side bets – the 6 card bonus side bet and the pair plus side bet. Three Card Poker - Hand Ranking; 3 card poker strategy differs from all other types of poker. Most poker hands are based on 5 cards regardless of how many are dealt. Even though 3 card poker uses the same poker hand hierarchy, it is difficult for many players, beginner and pro alike, to reconcile them into only 3 cards.
Brian Alspach
13 January 2000
Abstract:
The types of 3-card poker hands are
- straight flush
- 3-of-a-kind
- straight
- flush
- a pair
- high card
The total number of 3-card poker hands is .
A straight flush is completely determined once the smallest card in thestraight flush is known. There are 48 cards eligible to be the smallestcard in a straight flush. Hence, there are 48 straight flushes.
In forming a 3-of-a-kind hand, there are 13 choices for the rank and4 choices for the 3 cards of the given rank. This implies there are3-of-a-kind hands.
The ranks of the cards in a straight have the form x,x+1,x+2, wherex can be any of 12 ranks. There are then 4 choices for each card ofthe given ranks. This yields total choices. However,this count includes the straight flushes. Removing the 48 straightflushes leaves us with 720 straights.
To count the number of flushes, we obtain choicesfor 3 cards in the same suit. Of these, 12 are straight flushes whoseremoval leaves 274 flushes of a given suit. Multiplying by 4 produces1,096 flushes.
Now we count the number of hands with a pair. There are 13 choices forthe rank of the pair, and pairs of the chosen rank.The non-pair card can be any of the remaining 48. Thus, thereare 3-card hands with a single pair.
We could determine the number of high card hands by removing the handswhich have already been counted in one of the previous categories.Instead, let us count them independently and see if the numbers sumto 22,100 which will serve as a check on our arithmetic.
A high card hand has 3 distinct ranks, but does not allow ranks of theform x,x+1,x+2 as that would constitute a straight. Thus, there arepossible sets of ranks from which we remove the12 sets of the form .This leaves 274 sets of ranks.For a given set of ranks, there are 4 choices for each cardexcept we cannot choose all in the same suit. Hence, there are274(43-4) = 16,440 high card hands.
If we sum the preceding numbers, we obtain 22,100 and we can be confidentthe numbers are correct.
Here is a table summarizing the number of 3-card poker hands. Theprobability is the probability of having the hand dealt to you whendealt 3 cards.
| hand | number | Probability |
| straight flush | 48 | .0022 |
| 3-of-a-kind | 52 | .0024 |
| straight | 720 | .0326 |
| flush | 1,096 | .0496 |
| pair | 3,744 | .1694 |
| high card | 16,440 | .7439 |
last updated 13 January 2000
3 Card Poker Hands Ranked
Three card poker is always played with one deck of 52 cards. There are a total of 22,100 different ways that three cards can be dealt. Using a bit of math we can determine the probability of each hand. The table below shows the ways, probabilities and odds for each hand.
| Hand | Ways | Probability | Odds |
| Straight Flush | 48 | 0.002172 | 1 in 460 |
| 3 of a Kind | 52 | 0.002353 | 1 in 425 |
| Straight | 720 | 0.032579 | 1 in 30 |
| Flush | 1096 | 0.049593 | 1 in 20 |
| Pair | 3744 | 0.169412 | 1 in 5.9 |
| Nothing | 16440 | 0.743891 | 1 in 1.3 |
| Totals | 22,100 | 100.0000 |
Notes
3 Card Poker Hands Ranking
As you can see on the table above the straight in three card poker out ranks the flush. This is unique to the game as in every other poker game the flush always out ranks the straight. The reason for the change in rankings is the fact that there are more ways to make a flush with three cards than there are ways to make a straight.
3 Card Poker Hands
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