# Odds of Hypothetical Hands in Poker

Some friends and I were talking about playing Texas Hold 'Em according to some bizarre house rules that we brainstormed. Basically, the house rules allow for three new types of hands. In addition to

Pair
Two Pairs
Three of a Kind
Straight
Flush
Full House
Four of a Kind
Straight Flush
Royal Flush

our system allows for a “Color Flush,” which is to say, five cards of the same color (a mix of spades and clubs, or a mix of hearts and diamonds); “One of Each,” which is one card of each suit plus a kicker; and a “Royal Straight,” which is 10-J-Q-K-A, but not of the same suit.

What we can’t figure out is the rarity of each hand relative to the others. By my totally un-mathematic guesstimations, a “Color Flush” would beat a straight but not a regular flush. A “One of Each” would beat a flush but not a full house; and a “Royal Straight” would beat a regular straight but not a flush.

Can any mathematicians straighten out my math here?

I’m sure that someone will come along to help you with your question. I’ve often wondered about the “same color” option myself. I’m guessing it would fall between a pair and 2 pair in Texas Hold 'Em.

I just want to point out that there is no reason to distinguish between a “royal flush” and a “straight flush”, since one is just the highest type of the other. Same thing goes for a royal straight. It would be like trying to distinguish between a pair and a pair of Aces.

Any old straight flush is nine times more likely to show up than a royal flush. That sounds like a significant difference to me.

On to the custom hands:
[ul]
[li]Color flush, [sub]26[/sub]C[sub]5[/sub]/[sub]52[/sub]C[sub]5[/sub].[/li][li]One of each, 13[sup]4[/sup]*[sub]48[/sub]C[sub]1[/sub]/[sub]52[/sub]C[sub]5[/sub].[/li][li]Royal straight, 4[sup]5[/sup]/[sub]52[/sub]C[sub]5[/sub].[/li][/ul]

Someone else will have to provide the decimal values, as I don’t have access to my calculator right now.

But you also have to go back and rework all the old probabilities, because many hands that were pairs are now counted as color-flushes, and so on.

2.53%

52.74%

0.0394%

Values, of course, are not exact.

It is a difference, but it’s not significant. Or at least not significant in this context, which is trying to determine what hand beats another hand.

A straight flush to the ten beats a straight flush to the 8. A straight flush to the King beats a straight flush to the Jack.

In other words, as between two straight flush hands, the hand with the highest card is the winner.

Since a “Royal Flush” is simply a straight flush to the Ace, it’s meaningless to distinguish between the two. A straight flush to the Ace beats any other possible straight flush already; there’s no need to define it as a separate variety of hand. It already wins.

One thing to take into account is that with hold’em, you’re trying to make the best 5 card hand out of 7 possible cards, not 5. The original poker hand rankings were all calculated for 5 card hands; luckily, they work for 7 cards as well. They don’t necessarily work for other numbers. For example, in 3 card poker, a straight (6-7-8) is higher than a flush, because it is rarer.

Consider your “color flush.” With 7 cards to chose from, the only way to not have a color flush is if there are exactly 4 red and 3 black cards, or 4 black and 3 red cards. Any other distribution gives you this hand. The odds of a 4/3 distribution of colors is about 55%. Thus someone will hit a color-flush 45% of the time, making it even more likely than a lowly one pair.

One of each is even easier to get. The odds of getting a card of a particular suit are about 86.65%. .8665^4 = .56. Someone will make a “one of each” about 56% of the time.

I hope I did the above calculations correct–if not, someone will probably be in here shortly to point out my errors. No time to calculate a royal straight probability now–24 is almost on!

Color-flush seems to work well with, say, 3-card poker or 5-card stud. You’d also have to add an “All Black Straight” and “All Red Straight.”

This is starting to sound vaguely reminiscent of US-rules mah-jongg…

Why, oh why?

Please, whatever you do, don’t continue to call this game Poker after you’re done bastardizing it.

So, yay!

You’ve created two hands that fall somewhere between king high and a pair of deuces.

Again, why?

Ok, but let me introduce my ideas for the occasional 6 card hands, namely three pair, and two trips (more formally called at Full House Royale, and not to be confused with quads and a pair (the Full House Supreme)). These of couse would have to be ranked with 6 card flushes and straights.

Right, there is no need for a special name for that hand- it’s kinda like “Dead man’s hand” for AA88- there is a name for it, buit there is no need to place it in the poker hand hierarchy.

There are some historical hands for poker: Four-flush, skip-straight , Blaze, & “around-the-corner striaght”. A sample skip straight would be"2,4,6,8,10", a sample “round-the-corner” straight would be “K, A, 2, 3, 4” both of those hands rank just below a “real straight”. A “Blaze” is all face cards, it beats Two-pair.

It is fairly possible that if the OP went to this site

he could well find his hands listed by another name.

Keep it simple; make everything wild.

Unless I’m missing something, the probability of getting all red cards is [sub]26[/sub]C[sub]5[/sub]/[sub]52[/sub]C[sub]5[/sub], and of course the same is true for the probability of getting all black cards. The probability of getting one or the other of these cases is thus the sum of these two probabilities (2 * [sub]26[/sub]C[sub]5[/sub]/[sub]52[/sub]C[sub]5[/sub]) minus the probability of getting both simultaneously (zero).

[del]I don’t know if these are 100% accurate,[/del] but a while ago, I found some odds for the chances of getting xxx hand in 5 cards. (So draw and 5+n card games are different.)

``````
Combination:.......1 in # hands

High card:.........1:1
Pair:..............1:2
Two Pair:..........1:13
Three of a kind:...1:35
Straight:..........1:132
Flush:.............1:273
Full House:........1:586
Four of a kind:....1:3,914
Straight Flush:....1:64,794
Royal Flush:.......1:647,940

``````

…but, in checking some numbers, I get that, with a 52 card deck, there are 311,875,200 different hands (5251504948). Since only 4 of those hands are royal flushes, that gives a ratio of 1:77,968,800 chances of getting a RF. Hmmm… now I am curious. I wonder if those are the odds for the best 5 card hand out of 7. I may see if I can come up with an algorythm to run through all of the different combinations and simply count up the numbers… The plus side is, if I can do it, I could come up with some odds for the “invented” hands. (I am sure, however, that my stats teachers would weep to hear me attempt a calculated version.)

Your stats teachers would weep to learn that you completely missed the distinction between combinations and permutations. The number you calculated was for the latter (the number of different sequences of 5 items can be formed from a set of 52 items). The relevant number is for the former (the number of different subsets of 5 items can be formed from a set of 52 items). The difference is that the order of the 5 items counts for the former, but not the latter.

The number of different poker hands is actually 52! / ((52-5)! * 5!), which comes to 2,598,960. Your calculation is off by a factor of 5! because you overlooked the fact that it doesn’t matter which card in a poker hand is dealt first, second, etc.

Yeah, I figured that out sometime in between posting and now.
Rechecking the odds I posted, I now agree with the odds for a Royal Flush, but not for a Straight Flush. (But the difference is in how you define a “Straight Flush.” Since they include Royal Flush as a separate category, I would define “Straight Flush” to not include an Ace-High Straight Flush.) I may or may not run the program I was thinking of earlier.

The thing about poker hands is that they are unique in this sense, you cannot have two pair and have a straight. You cannot have a full house and have a flush. The only combintation hands are the straight flushes. These new hands create too many (IMO) more combination hands. Two-pair/all-color, one of each/fullhouse etc.

I’m assuming you’re talking about five card hands only,yes? Because with community cards you can have two pair and a straight with different combinations of the seven cards (e.g. JJ in the hole, QQT98 on the board). We’ll forego any discussion of Omaha and all the various screwball hands and draws that can result from having four hole cards.