The likelihoods of being dealt various poker hands are widely published (easily found on the internet). A more difficult problem is: what are the likelihoods of being dealt each poker hand, given a 54 card deck (52 card deck + 2 jokers).

The various hands of interest are:
1 pair
2 pair
3 of a kind
straight
flush
full house
4 of a kind
straight flush
5 of a kind

* Jokers can count as any rank card, in any suit.

hand      52 number 4card 2x upgrade 3Card upgrade 54 Total

5-of-a-kind -  0          0          26              0        52          78

straight flush 40       164        328           256     256         624

4-of-a-kind  624        13       4992             0     3744       9,360

full house 3,744          0       5616              0        0       9,360

flush         5,108     2696      5392          888       888   11,388 

straight    10,200   10332    20664        3840      3840  34,704

3-of-a-ki  54,912    2496   164736           52    13320  232,968

two pair  123,552    2808            0             0          0   123,552

pair     1,098,240  82368   339696       3744           0 1,437,936

high c  1,302,540 169848           0     13320           0 1,302,540

           2,598,960 270,725 541,450   22,100   22,100 3,162,510

 

I computed the 2,598,960 combinations for a standard 52 card pack which agrees with the standard table.  I then analysed the 4-card stems returning the actual results for pairs, 3 and 4 of a kind.  The remaining high cards were analysed as potential straights and flushes.  Since there are two jokers in play, I upgraded these hands to 2x their potential.

I repeated the process for 3-card stems and upgraded these in similar fashion.

{Edit: 3744 additional 1 pairs upgrade to 4 of a kind not full house as in previous.

The answer now agrees with web results http://www.durangobill.com/Poker_Probabilities_5_Cards.html

Other sites give different answers but this is because they change the ranking of the hands to adjust for the changes in frequencies.}

I am reasonably confident in the result, since all the figures were obtained separately by brute force and sum to the expected total number of combinations for 54 cards.

Edited on February 27, 2006, 11:35 am

Edited on February 27, 2006, 3:55 pm

Posted by goFish on 2006-02-27 11:25:19