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**17 members have voted**

The Royal Deck uses 100 cards which are the Σ of all cartesian products with 3 independent sets with elements of (Ace, King, Queen, Jack, Deuce) | (♤,♡,◇, ♧) | (Black, Red, Blue, Green, Yellow). Now I am going to be nice and provide the list of all possible hand rankings that can be derived from 5 or fewer cards (trust me that's a difficult task).

Your job is to put these hand rankings in order from least to greatest probability of occurring & then provide me with the hand ranking percentages for each of the 40 hand rankings + Fools Hand (which isn't ranked). Remember that the sum of all probabilities of all hand rankings is = 1, so I'm not asking for the probability of making a particular hand ranking or better which means if you have a pair but you also have doublet then you don't have a pair since the higher hand rank would be counted.

Start by just seeing if you could put all the hand ranking with 4 or fewer cards in the order form their probabilities if 4 cards are drawn. I will pay $100 to anyone who can do this for all the hand rankings if 5 cards are drawn & I'm leaving this challenge open for 30 days, that's how confident nobody can solve this question but me! Also don't forget that I solved the 4 card probabilities by hand with not even a calculator. I only used a computer to solve 1 thing for the 5 card probabilities and the amount of work it would take to write a program to solve this for you is much more than 30 days of work and if you are able to do that you should contact me and discuss terms on creating an app in which we can both become millionaires from this ;)

P.S. Please remove yourself from this form if you attempt to use calculus to solve this problem!

Does it matter which cards have to be the same suit to make a suited hand?

No, or I would have specified. Example a suited boat can be any combination of 2 & 3 suits like: A♤, A♡, A♤, K♡, K♤

Is a quint always a rainbow ranked hand?

Yes, every quint must also be a rainbow combination.

What is the difference between painted and colored?

Colored hands have < 5 cards with the same matching color, paint(ed)/(ing) hands have 5 cards with the same matching color.

What is the difference between suited & flush?

Suited hands have < 4 cards with the same matching suit, flush hands have 4 to 5 cards with the same matching suit.

Is there a 2-tone boat?

No, all 2-tones are doublets or doubles.

Symbol Meanings?

∆ = The combination of different elements. The delta symbol in logical analysis is referred to as the difference symbol.

∀= Any combination of characteristics. In predicate logic this symbol means "for all" or "for any".

🚫 = The absence of a combination of characteristics.

☆ = The color of a card. I chose the star symbol to represent different color or wavelengths of electromagnetism.

🄲 = The character or face value of a card.

♤ = The suit of a card

Quote:USpapergamesP.S. Please remove yourself from this form if you attempt to use calculus to solve this problem!

What is with all the hating on calculus?

Quote:WizardWhat is with all the hating on calculus?

No, this is to prove I'm better at combinatorial analysis for probabilities than every member of Wizard of Vegas. I value this type of mathematics over anything else since this is what I'm passionate about!

This is 5 ranks, 4 suits, 5 colors. 5*4*5 =100 cards. The only dififculty is in understanding the OP's terminology for defining hand categories.

I have already noticed that under the Non-Paired Hands the OP defines a category of hand that does not exist, namely "Rainbow Flush." Because there are only 5 ranks, any unpaired 5 card hand must, by definition, be a straight: AKQJ2. Therefore, any unpaired rainbow flush must necessarily be a rainbow straight flush , which is defined as a different hand category. Therefore, there are zero combinations for Rainbow Flush.

Also zero combinations for these non-paired hand categories defined by OP:

Painting (duplicates Painted Straight)

Rainbow (duplicates Rainbow Straight).

Also, the Fool's Hand has only 4 cards. I don't understand how that makes sense.

EDIT: I now see that OP defines the word "Flush" as meaning that 4 or 5 cards are of the same suit. So a five card hand with 4 spades and one diamond is a "flush"??????

It didn't take long for Mr Arrogant to get banned.

Quote:gordonm888I could certainly do this. IMO, there are maybe more than two dozen. active forum members who could do this.

This is 5 ranks, 4 suits, 5 colors. 5*4*5 =100 cards. The only dififculty is in understanding the OP's terminology for defining hand categories.

I have already noticed that under the Non-Paired Hands the OP defines a category of hand that does not exist, namely "Rainbow Flush." Because there are only 5 ranks, any unpaired 5 card hand must, by definition, be a straight: AKQJ2. Therefore, any unpaired rainbow flush must necessarily be a rainbow straight flush , which is defined as a different hand category. Therefore, there are zero combinations for Rainbow Flush.

Also zero combinations for these non-paired hand categories defined by OP:

Painting (duplicates Painted Straight)

Rainbow (duplicates Rainbow Straight).

Also, the Fool's Hand has only 4 cards. I don't understand how that makes sense.

Yes, making sense of it is the only hard part.

1. Where does a "Painting" (1-color hand) or a "Rainbow" (5-color hand) rank? Remember that every hand that is not a straight has at least a pair.

2. What, exactly, is a "Fool's Hand"? Does AKQQJ count as one?

3. Should the fifth card in a "two-tone flush" be the same suit as the other four?

4: In a "suited two-tone" or a "two-tone," can one of the pairs be two cards one suit and the other pair two cards of another?

5: What, exactly, is a "suited quint"?

6: Where is a 5 of a Kind that is neither a flush nor a rainbow?