# Motivation

Have you ever played Texas Hold 'em and wondered, "Why do my cards always suck? I often only have one pair, and sometimes it's already on the table and we're left comparing kickers." Indeed, the distribution of hands in Texas Hold 'em is quite skewed towards the bottom:

Keep in mind that these are the best five-card hands out of seven randomly chosen cards. Indeed, most hands do suck; only 15% of the time is there something better than two pair.

# Introducing: Red Seven Poker

The game of Red Seven Poker is played with a 49-card deck with seven suits (the seven colors of the rainbow) and seven numbers (1 through 7). The deck comes as a part of another game, *Red Seven* by Carl Chudyk and Chris Cieslik, which is available from Asmadi Games on Amazon.

The structure of Red Seven Poker is the same as regular Texas Hold 'em. Each player has two secret cards to themselves and there are (eventually) five cards available to everyone. At the end of all of the betting, if more than one player is left, the player left with the best five-card hand out of the seven cards available to them wins the pot. (For full Texas Hold 'em betting rules, follow the link above.)

# Ranking Hands

Here is the short reference version of the hand rankings Red Seven Poker.

- Royal Flush
- Straight Flush
- Flush / Five of a Kind
- Double Straight
- Full House Straight
- Four of a Kind
- Double Two Pair
- Double Three of a Kind
- Rainbow Straight
- Rainbow
- Straight
- Full House
- Three of a Kind
- Two Pair

Notice that there are many more possible hands than in regular poker, which adds variety. As we'll see later, many of those hands are also more common.

# General Rules and Tiebreakers

- The colors in Red Seven (unlike suits in regular poker) are ordered: Red is the highest color, then Orange, Yellow, Green, Blue, Indigo and Violet (commonly memorized as ROYGBIV). Fortunately, this reference guide comes with
*Red Seven*.

- Since the deck is symmetric, many of these hands can be achieved in two different ways: Using colors, and using numbers. For instance, this symmetry implies that a flush (five of one color) and five of a kind (five of one number) are just as common as each other, and therefore they are assigned the same rank.
- If two hands have the same rank, then ties are broken according to the tiebreak rules of the game
*Red Seven*: You compare the highest card out of the five, then the second-highest, and so on. This is where the symmetry breaks: Sevens are all higher than sixes, which are higher than fives, and so on. For cards with the same number, red is the highest and violet is the lowest. - The only exceptions to this rule are hands that treat some of their cards differently from others: For Four of a Kind and Double Two Pair, first the four cards are compared, then the extra (kicker) card. Similarly, for Three of a Kind, Full House and Full House Straight, first the set of three cards is compared, then the remaining two.

# Hand Explanations and Examples

**Royal Flush**. A*flush*is five cards of the same color or number. For the flush to be considered*royal*, it must be the highest five cards of the set: Either 7/6/5/4/3 or R/O/Y/G/B. Here are two examples of royal flushes:

The highest possible hand in the game is therefore the 7's royal flush.**Straight Flush**. A*flush*is five cards of the same color or number. A*straight*is five cards with consecutive values, either numbers (3-7, 2-6 or 1-5) or colors (R-B, O-I, or Y-V). A*straight flush*is a straight in color and a flush in number or the other way around. Here are two examples:

As with regular poker, all royal flushes are also straight flushes, but royal flushes are considered higher than all straight flushes.**Flush**. A*flush*is five cards of the same color or number. Here are two examples:

Of course, all straight flushes are also flushes, but those all rank higher.**Double Straight**. A*straight*is five cards with consecutive values, either numbers (3-7, 2-6 or 1-5) or colors (R-B, O-I, or Y-V). A*double straight*is five cards which form a straight in both colors and numbers. Notably, order of the cards in the two straights does not need to be the same; it just needs to be the same five cards. Here is one example, depicted as a straight in two different ways:

**Full House Straight**. A*full house*consists of three cards of one color and two cards of a second color, or alternatively three cards of one number and two cards of another number. A*straight*is five cards with consecutive values, either numbers (3-7, 2-6 or 1-5) or colors (R-B, O-I, or Y-V). A*full house straight*is therefore a set of five cards which form a straight by colors and a full house by numbers or a straight by numbers and a full house by colors. Here is one example of the latter:

**Four of a Kind**. Because of the deck's symmetry, we will say that a*kind*can refer to either a color or a number. So*four of a kind*can refer to either four cards of the same number or four cards of the same color:

**Double Two Pair**. This hand is one of the new additions.*Two pair*refers to four cards, two of which are one color and two of which are another color, or two of which are one number and two of which are another number. Just like Double Straight,*double two pair*refers to four cards which form two pair in both senses: The four cards are all one of two colors and one of two numbers. Like Four of a Kind, the fifth card can be anything. Here is one example, depicted both ways:

**Double Three of a Kind**. This hand is another new addition.*Three of a kind*refers to three cards that are the same color or the same number.*Double three of a kind*refers to*five*cards, three of which are the same color and three of which are the same number. Of course, those threes of a kind must share one card in common, the*anchor*. Here is one example, seen both ways:

**Rainbow Straight**. A*rainbow*is five cards, no pair of whom share a color or number. A*straight*is five cards with consecutive values, either numbers (3-7, 2-6 or 1-5) or colors (R-B, O-I, or Y-V). So a*rainbow straight*is simply both a rainbow and a straight. Here are two examples:

All double straights are rainbow straights as well, but rank higher.**Rainbow**. A*rainbow*is five cards, no pair of whom share a color or number. Here is an example:

**Straight**. A*straight*is five cards with consecutive values, either numbers (3-7, 2-6 or 1-5) or colors (R-B, O-I, or Y-V). Here are two examples:

**Full House**. A*full house*consists of three cards of one color and two cards of a second color, or alternatively three cards of one number and two cards of another number. Here are two examples:

**Three of a Kind**.*Three of a kind*refers to three cards that are the same color or the same number. Here are two examples:

**Two Pair**.*Two pair*refers to four cards, two of which are one color and two of which are another color, or two of which are one number and two of which are another number. Here are two examples:

# Frequencies

So Red Seven Poker introduces more possible hands, but that doesn't help if they're all super rare. What is the frequency of getting each hand?

As you can see, there is a lot more variety than in original poker. A full 86% of hands are better than two pair, which is now the lowest-ranked hand. The median hand is a straight, and there is plenty of competition higher. Anecdotally, I've been burned by betting high on Double Two Pair only to be beaten by a Four of a Kind or a Double Straight.

We can make this judgment a bit more rigorous by calculating the entropy of these two distributions. Indeed, original poker hands have an entropy of 2.2 bits, while Red Seven Poker has entropy of 3.2 bits. That might not seem like much, but that one additional bit of entropy corresponds to roughly doubling the number of potential hand ranks you could get.

# Frequently Asked Questions

**How did you decide the order of the ranks? Why is Double Three of a Kind ranked lower than Double Two Pair, even though it's less common?**

As another example, Three of a Kind is less commonly the best in the hand than Full House. But no one would suggest that Three of a Kind should beat Full House: Every time you have a full house, you also have three of a kind! More generally, any 7-card hands will have 5-card subsets with several different ranks, and we can count how often a particular rank shows up as one of those. If we do that, this is the result:

As you can hopefully see, the hands are in the appropriate order. Double Three of a Kind is actually quite a bit more common than Double Two Pair counted inclusively (9.0% versus 7.1%). This is both because there are 7-card hands that include both Double Three of a Kind and Double Two Pair and because there are more 7-card hands with both Double Three of a Kind and Four of a Kind than there are with both Double Two Pair and Four of a Kind.**How can I easily identify a rainbow?**

If you can spot one, it's easy to check that all the colors and numbers are different. Perhaps it's trickier to see if you don't have one: If there are three separate pairs of cards that share an attribute, then there isn't a rainbow.**How can I easily identify a straight in colors?**

This is also tricky. Obviously, the five cards must be five different colors. After that, I find it easiest to look at which colors are not represented. For a straight, the missing colors must be Red+Orange, Red+Violet, or Indigo+Violet. Alternatively, you can first check that you have Yellow, Green and Blue (which are part of any straight in colors).**How should I think about a given rank? Which ranks are good?**

Of course, it depends on what cards are on the table; if there are three of one color or number, for instance, Four of a Kind becomes more likely. But generally the inclusive ranks can be grouped into three categories: Common (Two Pair through Rainbow), Uncommon (Rainbow Straight through Four of a Kind) and Rare (Full House Straight through Royal Flush). All of the common ones happen on at least a quarter of hands; the Uncommon hands occur about 6-10% of the time, and Rare hands occur less than 3% of the time. Stated differently, you should get an Uncommon hand about every 3 deals and a Rare hand about every 30 deals.**How did you compute all of these frequencies? By hand?**

Good question! There are around 86 million possible 7-card hands from a 49-card deck, and the short answer is that I checked all of them with a Mathematica script. It took some effort to cut down the time the script ran, but in the end, it only took a few hours on my laptop. I then double-checked the handful I could compute by hand.