I am playing around with creating a new card game, similar to poker, and I am really struggling with figuring out the math behind calculating the probability of certain hands. Heres the overview of what I am trying to do.
- There are only 3 types of hands that are playable - Straight, Flush, Straight Flush.
- Unlike normal poker, a 3, 4 or 5 card straight/flush/straight flush can be played.
- Player hands are 5 cards total.
What I originally wanted to do was just put the following list of possible hands in the order of probability:
- Three Card Straight;
- Three Card Flush;
- Three Card Straight Flush;
- Four Card Straight;
- Four Card Flush;
- Four Card Straight Flush;
- Five Card Straight;
- Five Card Flush;
- Five Card Straight Flush;
I have this assumption that the probability of a 3 card Straight Flush is lower than say a 4 card Flush - but I'd like to understand the math behind proving that assumption right/wrong..
Now to the part that is melting my brain! The information on calculating the probability for the normal 5 card poker hands is well documented, so I can get the 5 card hand probabilities easily. Using Flush as an example, and not discounting the possible straight flushes just to make it easier:
Total Number of possible hands from a deck of 52 cards, with 5 card hands: 2,598,960
Five Card Flush Probability:
( C(13,5) x C(4,1) = 5148 (total number of 5 card flushes)
5148 / 2598960 = 0.1981%
So I tried to do the same for a 4 card flush, I thought it would be:
( C(13,4) x C(4,1) ) = 2860
2860 / 2598960 = 0.1100%
But clearly this isn't right as it suggests getting a 5 card flush, is more likely than a 4 card flush!
Any help would be greatly appreciated!