Question - Frequency of Hands...
by TheAdviser | Published Jan 16, 2014
One of the things that I have always found interesting about Poker is how the hand rankings compare to the actual frequency of the hands. I guess historically, someone figured out which hands are the most and least common, and decided that those which are the least common should be considered to be the most powerful, and thus be considered winners over the less common hands. We take that for granted now, but someone, somewhere, decided on these rules.
Most every player knows which hands beat which other hands, but I wonder how many really have a feel for the frequency of each hand. One thing that I noticed is that relative (or adjacent) probabilities of being dealt each hand are not uniform. For example, with 5 cards, it is about 15 times more likely to get quads than it is to get a straight flush, but it is only 6 times as likely to get a boat as it is to get quads. And it is only 1.4 times more likely to have a flush than to get a boat.
Does anyone know of a good discussion somewhere, of this non-uniform frequency in the hand rankings? And how it might relate to strategy?
It seems to me that you were trying to devise a poker strategy, you would want to take into account both the rank (Power) and the commonality (Frequency) of the various hands. Shouldn't a player try for those hands which have the best combination of Power and Frequency? But what is that combination?
Taking any two cards to the flop, for a total of 5 cards, gives the following probabilities. The Multiplier column shows how much more frequent each hand is than the higher ranked hand coming before it. (I'm mainly interested in the state of the game on the flop, as that is where most hands are truly defined, informed decisions can be made.)
HAND___________ COMBINATIONS________ MULTIPLIER
Straight-Flush_____ 40___________________ na
Quads___________ 624_________________ 15.6
Boat_____________ 3,744________________ 6.0
Flush____________ 5,108________________ 1.4
Straight__________ 10,200________________ 2.0
Trips____________ 54,912________________ 5.4
Two-Pair_________ 123,552_______________ 2.3
Pair_____________ 1,098,240_____________ 8.9
Nothing__________ 1,302,540_____________ 1.2
Total 5 Card Hands_ 2,598,960_____________ 2.0
(much more useful in chart form, but sadly I cannot post a chart on CardPlayer)
I don't know exactly how to Apply this information in a poker strategy, but we can make some interesting observations...
- Quads are over 15 times as likely to appear as a Straight-Flush. That's a lot!
- A Boat is 6 times as likely as Quads are. Also seems like a significant amount.
- But a Flush is only 1.4 times as likely as a Boat. Boat / Flush is the tightest pairing in frequency between any two ranks.
- A Boat, Flush, and Straight all happen with relatively comparable frequency. I think those can all be reasonably grouped together. In fact, a Straight only happens 2.7 times as often as a Boat, despite there being two full steps in rank between them.
- Finally a Pair is 9 times as likely as Two-Pair. This also seems like a big step between ranks.
And one observation from the basic rules of poker itself...
- There is no bonus for blowing your opponent out of the water. If your opponent has a simple Pair, you only need a higher Pair or Two-Pair to take down the pot. Having a Straight-Flush does you no extra good.
Is there any practical value to this information? Can any useful strategy tips come from this irregularity in the frequency between the hand ranks? And is there a mathematical way of incorporating both Rank and Frequency into a single number for comparison purposes?