Mathematics for Card Ranking *Help*

I don't feel like math tonight, but another game that I own (Tiny Epic Western - really not that great of a game, btw), did this same ordering for three-card poker hands.

That is, if you can discard and draw (like in draw poker), then if 3-of-a-kind beats a straight, you're more likely to go for that, vs. if it doesn't. That skews the actual results.

Are you asking for the likelihood of being DEALT one of these hands? Because that's comparitively easier.

6 suits, 9 cards per suit. Should be pretty straightforward to calculate:

7 ways to make a straight flush in one suit, so 7 * 6 = 42 total straight flushes.

6 suits give 6-choose-3 ways to make three-of-a-kind, which is 6!/(3!*3!), which is 20. 9 cards, so 180 total threes-of-a-kind.

For flushes, you have 9-choose-3 ways to get a flush in one suit, which is 9!/(6!*3!), which is 84. 6 suits, so 504 total flushes. Subtract the straight flushes, so it's really 462.

For straights, you have 7 ways to make a straight, with 6 choices for the suit of each card, so 7*6*6*6 = 1512. Again ignore the straight flushes, so 1470.

For a pair, there are 6-choose-2 = 6!/(4!*2!) = 15 ways to make each pair. The third card can be anything OTHER than that paired card (because that would be 3oak), which gives 8 choices. And there are 9 card types, so 15 * 8 * 9 = 1080 pairs.

Which may come as a slight surprise :).

There are 54*53*52 = 148,824 possible draws, which leaves a high card as the overwhelming favorite to get. So overwhelming that I hope this game does have a draw rule or more cards dealt later or something.

Oh, well that's totally different, so.. you're on your own for that analysis!

And the joker too screws everything up.

I'm not sure if this calculator would help you, but I'll throw it out there any way.

http://www.unseelie.org/cgi-bin/cardco.cgi

You know, perhaps the first hand should be calculated. But everything beyond that is a gamble. Since players can choose or not. Maybe, just maybe, you can calculate the replacement of cards when trying to aim for something better. But that would mean that you build a tree of possibilities.

I have done this before. But let's say that a tree with many branches that can't "collide" or is depending on permutations. Is very hard to achieve. A lot of work goes in it. So, the best way to calculate your odds is to have a program work it out.

I can't program such thing. But maybe someone else here will. I asked for help before too when facing probabilities of the "usage" of my cards depending on strict rules.

This is my suggested plan:
- You make the rules as clear as possible for how cards are distributed.
- The programmer turns it into a counting program. For first hand.
- And after that, replacements of the worst cards. So you should get multiple columns; first hand, 1 card replaced, 2 cards replaced etc.
- You need to be clear on which cards will be replaced first.
- You need to be clear on if cards are going to be replaced any way.

With the last part of this plan. This is actually only doable when the first hand is calculated. After getting the results of the first hand. You could look at the first replacement, if certain combinations rise "practically" to be bothered with.
If the rise isn't practically. You are stuck with the first hand. Which is an easy way out, but a correct one.
If the rise is practically. I guess the problem just got more difficult. But it would show that replacing a card is very worthwhile the players time.

It is kinda how I did things for my game. But I had the luck that the cards depended on the situation of the game itself. So I simply had to see for the entire game. Which might be an option for you too. But you need a good program. This can't be done with paper.