Hi guys.

I run a poker platform in Asia and I have been searching all over the internet for the below information to no avail, I will give some background information to what I am trying to do so that it is not just a what is the answer to 1+1 then someone answers 2 and I say thank you.

Firstly, I am no maths major but I understand the basics certainly not at a university level, I majored in finance.

I am trying to implement a Jackpot system NOT a bad beat jackpot system.

Basically, in a round of holdem poker whenever the pot reaches X big blinds (BB) the system will take Y BB from the pot and place it into the jackpot pool.

I have now encountered the following:

1) I am having trouble finding the probability of any hand hitting (must use BOTH HOLE cards, in the case of four of a kind must use BOTH HOLE cards and needs to be a pocket pair)
- royal flush
- straight flush (not including royal flush)
- four of a kind

There are many readily available articles and websites detailing probabilities for 5 card poker and 7 card poker (holdem) but none with odds relating to what I wrote above making use of BOTH HOLE cards.

The reason why I wish to know these probabilities is so that I can estimate dealing how many hands of holdem I can expect each of the above 3 scenarios to happen then I can estimate how much jackpot rake I have taken.

2) Our jackpot system does not contain bad beat jackpot as those are too rare and not enticing for players, rather the jackpot pool that we are implementing simply pays out a % of the pool eg,

- royal flush 50%
- straight flush (not including royal flush) 25%
- four of a kind 5%

My second question is how should I setup these percentages to ensure that over the long run (statistically) that the jackpot grows over time, ie, the max values for each type of hand (quads, straight flush, royal flush) that I can payout so that the pool keeps growing. I would not want the jackpot pool to become smaller and smaller because then it would not be enticing over a long period of time.

We are thinking about taking and placing into the jackpot pool：

1/2 BB when if the pot is between 15BB and 30BB
1 BB when if the pot reaches 30BB or more

As you can see the maximum we can take from a pot is 1 BB.

Would greatly appreciate some help on the 2 questions I have above.

Thanks everyone.

We have had a few recent threads on this topic. I will summarize the results from those threads. Many members of this forum have contributed to these results. As per usual, these results assume all players go to showdown on all deals.


(1) Quads of any rank in NLHE using a pocket pair (one player)

= 13*C(4,2)*C(48,3)*C(2,2) / C(52,7)*C(7,5)

= 1,349,088 / 2,809,475,760

= 0.000480192

which is approx once every 2,082 deals.


(2) Royal Flush in NLHE using both hole cards (one player)

= 4*C(5,2)*C(47,2)*C(3,3) / C(52,7)*C(7,5)

= 43,240 / 2,809,475,760

= 0.000015391

which is once every 64,974 deals.


(3) Straight Flush (excluding royal flush) in NLHE using both hole cards (one player)

= 4*9*C(5,2)*C(46,2)*C(3,3) / C(52,7)*C(7,5)

= 372,600 / 2,809,475,760

= 0.000132623

which is approx once every 7,540 deals.


I would highly recommend that you wait for confirmation of these results before utilizing them.

Also, the above results are for one player at a NLHE table. For N players at a NLHE table, you would need to multiply the above probabilities by N (which are very very close to the actual probabilities).

I/we will turn our attention to your second question in a subsequent post.

Thank you very much - very very helpful

I use a slightly different method.

Quads:

Pr(pocket pair) = 1/17. Assume w.l.o.g. black aces.
conditional Pr(Ah on board) = 5/50
conditional Pr(Ad on board) = 4/49

Probability = 20 / 41,650 = 1 / 2082.5

Royal:

Pr(royal cards) = 20*4 / 52*51 = 20 / 663. Assume w.l.o.g. AsKs.
conditional Pr(Qs on board) = 5 / 50
conditional Pr(Js on board) = 4 / 49
conditional Pr(Ts on board) = 3 / 48

Probability = 1 / 64,974

Our solutions match up for quads and royals, computed 2 different ways. Seems unlikely we are both wrong.

There is no such quick / dirty / easy way to figure out the probability of a straight flush so I'm lazy enough to just trust you

Thanks !!