Calculating probabilities for 3, 4, 5 card Straight/Flush/Straight Flush

Question

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:

1. Three Card Straight;
2. Three Card Flush;
3. Three Card Straight Flush;
4. Four Card Straight;
5. Four Card Flush;
6. Four Card Straight Flush;
7. Five Card Straight;
8. Five Card Flush;
9. 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)

Probability: 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

Probability: 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!

Answer 1

One at a time and you are way better off on stats.stackexchange.com
I am a mathematician but not a statistician
Straight is harder ways-to-make-a-straight-in-poker

On flush if you are playing 2 plus board best 5 you need to look at 7
Define game rules

( C(13,4) x C(4,1) ) = 2860 is wrong
the fifth card is 52-4
C(13,4) x C(4,1) x (52-4)

The above includes 5 card flush
For 4 only I think it it would be
C(13,4) x C(4,1) x 39

I can tell you right now that you are going to get some whacked out order that flop from one to the other and is not going to be embraced.