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I developed a Video Poker software. Its working fine "Jacks Or Better"

Now i want to improve it with statistics.I want the software to suggest which cards to hold to get the best possibility of a gain.

I know that holding $5$ cards in different way create $32$ scenarios.

What i want is to calculate the probability for each one of these $32$ scenarios :

  1. Royal Flush
  2. Straight Flush
  3. Four Of A Kind
  4. Full House
  5. Flush
  6. Straight
  7. Three Of A Kind
  8. Two Pairs
  9. Jacks Or Better

I got the solution for Jacks or Better when nothing is hold ...

its: $13C1 * 4C2 * 12C3 * (4C1)^3 * \frac4{13}$

But whats the probability when my hand is

$JS - 2C - 6D - 7H - 10H$ and i hold $JS$.

Is it possible to develop a formula for each of $9$ ( Special Hands ) for a given scenario ?

When nothing is selected is easy. But when you hold 1 , 2 or more cards depending what you want to achieve its very difficult for me ...

Please , i am struggling with it for a week and got formulas just when holding nothing ...

Thanks in advance ...

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  • $\begingroup$ You do know that in actual video poker machines, the odds are altered by non-uniform dealing, such that the casino can set the house percentage take wherever they want (within limits) just as on slot machines. $\endgroup$
    – Mark Fischler
    Commented Mar 31, 2017 at 23:33
  • $\begingroup$ When nothing is selected (meaning, I assume, when the player discards all five cards and draws a ful new hand) the calculation is still not easy, because unless your machine cheats by re-using the discarded cards, the actual odds very slightly change depending on which five cards were discarded. $\endgroup$
    – Mark Fischler
    Commented Mar 31, 2017 at 23:49

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