Can We Let N Choose K?

In poker, only the class of hands are ranked correctly.  However, the hands within the class are ranked completely backward, and wow, poker would be so much different if N really did choose K.

What does this mean?  In short, a Seven High Flush would beat an Ace high Flush.  Two Pair of Threes and Deuces would beat Aces and Sevens.  But why?

Let’s take a look at something.  There is exactly one way to draw a Royal Flush (AKQJT) and it’s frequency is four (one of each of the suits).  If you draw 5 cards at random from a deck of 52, you will draw a Royal Flush only once in 649,739 attempts.  How do we get to that number?  We divide the number of Royal Flushes in the deck, 4, by the total number of possible hands.  It’s easy to see that there are only four Royal Flushes, but how do we arrive at the total number of five card hands.  Well, we let N choose K where N is the number of cards in the deck, and K is the number of cards used in the poker hand.  We will express this function here as:  (N ~ K).  The expression is written below and can be entered on a scientific calculator by using the nCr function.

(N ~ K) = (N!)/[K!(N-K)!]

When we plug in the number, we get:  (52!)/(5!(47)!).  For those of you that don’t know, the ‘!’ in mathematics is pronounced “factorial.”  This is simply the number multiplied by one less than itself and repeated until it is multiplied by 1.  Example:  4! equals 4 times 3 times 2 times 1 (4 x 3 x 2 x 1 = 24).  Back to topic–the total number of possible hands is:  (52 ~ 5), 52 cards in the deck and we are using 5 at a time.  This gives us 2,598,960 total hands.  Four of those hands are a Royal Flush, so we make a Royal Flush 4 out of 2,598,960 times which reduces down to 1 in 649,739 attempts.

Phew, the hard part is over.  I hope you made it this far, because the rest is a breeze.  Now that we know exactly how many possible hands there are, and how to calculate the probability of getting a hand, we will continue.  It’s easy to see how many Royal Flushes there are:  exactly one way to make a Royal times 4 different suits.  The rank of poker hands are as follows:  Royal Flush, Straight Flush, Four of a Kind, Full House, Flush, Straight, Three of a Kind, Two Pair, One Pair and High Card.  I will not bore you with how to make each one, instead I am going to tell you that these classes are ranked correctly.  It is harder to make a Royal Flush than a Straight Flush.  It is harder to make a Straight Flush than Four of a Kind.  You get the picture.  However, again, the rank within the classes are flawed.  Let’s jump to a Flush.

There are 1,277 ways to make a flush.  Multiply that by four suits and we get 5,108 different flushes.  Remember, there was one way to make a Royal Flush times four suits, so there are four Royal Flushes.  In poker, an Ace high flush beats a King high flush, beats a Queen high flush, etc etc.  However, an Ace high flush is the easiest flush to make!  The hardest? A Seven High Flush.  This is why:

To make a Royal Flush, you must draw AKQJT.  To make a Seven High flush, you must draw 23457 or 23467 or 24567 and that’s it!  Remember, 34567 is a Straight Flush.  So a 7-high flush is exactly thrice as likely as a Royal Flush.  The odds of making a 7-high flush are:  12 out of 2,598,960 which reduces to 1 in 216,580. To make an 8-high flush you must draw:  23458, 23468, 23478, 23568, 23578, 34568, 34578.  Thus, an 8-high flush is the second most difficult flush to make.  So on and so on, and an Ace-high flush is the easiest flush to make, but it beats all other flushes.

The same can be said for Two Pair hands.  It is much easier to make Aces Up than Threes Up, however, Aces Up beats all other Two Pair hands.  And High Cards hands??  Well, Ace High is MUCH easier to make than 7 High, just ask any Low Ball player.  So, the rank of the class of hands are correct, however, the rankings within the class are completely backward.  This could be the derivative of Low Ball games, where the Lowest hand wins.  After all, the lowest ranked hands within a class are the most difficult to make.  The rules of poker won’t change, we have to have some order!  Tie breakers within a class are determined by the highest ranked card and that’s just how it’s gonna be!  So, by all means, draw to the lower flush.  While you let N choose K, I’ll sit quietly and stack your chips!


Giveaway Question:  How many hours did we play on Labor Day Saturday?