Computing Probability of Starting Hands

Hi everyone,

Is there any resource that will compute how often each starting hand will be dealt in video poker, based on a particular strategy chart? I'm not referring to just the standard hands from royal flush down to 1 pair down to high card. Rather, I'm looking for something that would give me a breakdown of each of the hands on a VP strategy chart (like the ones listed on Wizard of Odds, I would link the specific one I'm interested in, but it seems the forums won't let me post a link. I'm assuming it's a spam protection until I've accumulated X number of posts, and not just general user error.)

I have a basic knowledge of combinations and permutations, and some modest Excel skill. However, I start to melt down when computing probabilities for hands lower down the list. For example, 4 to an outside straight (the 12th item on that strategy list). I can compute the odds of being dealt one easily enough. But then I need to account for not getting a straight flush, not getting 4 to an outside straight flush, not getting 4 to a gutshot straight flush, not getting 4 to a flush, and, for some of the possible 4-straights, 3 to a royal. Given enough time, I could probably muddle through and get it, but then it's not like it's going to get any better the further down the list I go. The possibility for errors is nearly limitless for me.

So, I'm hoping someone here knows of some tool that would compute the probability of each starting hand, accounting for the hands higher on the list. Much thanks!

This is a good question that I've often wondered. Obviously, pat hands can be looked up but some are harder to figure. If you are asking about a specific game and pay table, just list it. Someone here will be able to help.

Maybe if there is ever an update to the strategy makers, another column for the probability of each hold can be listed as well.

Aces or better:
Royal: 500
Straight Flush: 106.25 (yes, it’s really fractional due to getting better odds as you increase your bet, this represents the max bet)
Quads: 25
Full House: 8
Flush: 5
Straight: 5
Trips: 3
2 Pair: 2
Aces: 1

I still can't post links, or would include the link to the strategy for it. Maybe next post? ;)

Quote: jek187

Aces or better:
Royal: 500
Straight Flush: 106.25 (yes, it’s really fractional due to getting better odds as you increase your bet, this represents the max bet)
Quads: 25
Full House: 8
Flush: 5
Straight: 5
Trips: 3
2 Pair: 2
Aces: 1

I still can't post links, or would include the link to the strategy for it. Maybe next post? ;)

You need 10 posts before you can post a link. Somebody should be able to figure this out or point you in the right direction

Ah, thanks. For now, I'll just leave this here:
/games/video-poker/strategy/a-1-b-95-c-1-d-0-d-1-d-2-d-3-d-5-d-5-d-8-d-25-d-25-d-25-d-106.25-d-500/

If you put the wizard's site in at the beginning, that'll get you there.

If I am reading a request right, here is a table of all possible hands:

Quote: ThatDonGuy

If I am reading a request right, here is a table of all possible hands:

Click Here

"Ranks" is how many different sets of ranks exist for that hand. For example, for the first one, the four of a kind can be any of the 13 ranks, and the single can be any of the 12 remaining ranks; 13 x 12 = 156.

"Suits" is how many different sets of suits exist for each set of ranks. For example, for the second one (the full house where both suits in the pair are also in the three), there can be four sets of three suits for the three of a kind, and for each set, three pairs of suits for the pair; 4 x 3 = 12.

Hand	Ranks	Suits
As Ah Ac Ad Ks	156	4
As Ah Ac Ks Kh	156	12
As Ah Ac Ks Kd	156	12
As Ah Ac Ks Qs	858	12
As Ah Ac Kd Qd	858	4
As Ah Ac Ks Qh	858	24
As Ah Ac Kd Qs	858	12
As Ah Ac Ks Qd	858	12
As Ah Ks Kh Qs	858	12
As Ah Ks Kh Qc	858	12
As Ah Ks Kc Qs	858	24
As Ah Ks Kc Qc	858	24
As Ah Ks Kc Qh	858	24
As Ah Ks Kc Qd	858	24
As Ah Kc Kd Qs	858	12
As Ah Kc Kd Qc	858	12
As Ah Ks Qs Js	2860	12
As Ah Kc Qc Jc	2860	12
As Ah Ks Qs Jh	2860	12
As Ah Ks Qh Js	2860	12
As Ah Kh Qs Js	2860	12
As Ah Ks Qs Jc	2860	24
As Ah Ks Qc Js	2860	24
As Ah Kc Qs Js	2860	24
As Ah Kc Qc Js	2860	24
As Ah Kc Qs Jc	2860	24
As Ah Ks Qc Jc	2860	24
As Ah Kc Qc Jd	2860	12
As Ah Kc Qd Jc	2860	12
As Ah Kc Qd Jd	2860	12
As Ah Ks Qh Jc	2860	24
As Ah Ks Qc Jh	2860	24
As Ah Kc Qs Jh	2860	24
As Ah Ks Qc Jd	2860	24
As Ah Kc Qs Jd	2860	24
As Ah Kc Qd Js	2860	24
As Ks Qs Js 2s	1287	4
As Ks Qs Js 2h	1287	12
As Ks Qs Jh 2s	1287	12
As Ks Qh Js 2s	1287	12
As Kh Qs Js 2s	1287	12
Ah Ks Qs Js 2s	1287	12
As Ks Qs Jh 2h	1287	12
As Ks Qh Js 2h	1287	12
As Ks Qh Jh 2s	1287	12
As Kh Qs Js 2h	1287	12
As Kh Qs Jh 2s	1287	12
As Kh Qh Js 2s	1287	12
Ah Ks Qs Js 2h	1287	12
Ah Ks Qs Jh 2s	1287	12
Ah Ks Qh Js 2s	1287	12
Ah Kh Qs Js 2s	1287	12
As Ks Qs Jh 2c	1287	24
As Ks Qh Js 2c	1287	24
As Ks Qh Jc 2s	1287	24
As Kh Qs Js 2c	1287	24
As Kh Qs Jc 2s	1287	24
As Kh Qc Js 2s	1287	24
Ah Ks Qs Js 2c	1287	24
Ah Ks Qs Jc 2s	1287	24
Ah Ks Qc Js 2s	1287	24
Ah Kc Qs Js 2s	1287	24
As Ks Qh Jh 2c	1287	24
As Ks Qh Jc 2h	1287	24
As Ks Qc Jh 2h	1287	24
As Kc Qs Jh 2h	1287	24
Ac Ks Qs Jh 2h	1287	24
As Kh Qs Jh 2c	1287	24
As Kh Qs Jc 2h	1287	24
As Kh Qc Js 2h	1287	24
As Kc Qh Js 2h	1287	24
Ac Ks Qh Js 2h	1287	24
As Kh Qh Js 2c	1287	24
As Kh Qh Jc 2s	1287	24
As Kh Qc Jh 2s	1287	24
As Kc Qh Jh 2s	1287	24
Ac Ks Qh Jh 2s	1287	24
As Ks Qh Jc 2d	1287	24
As Kh Qs Jc 2d	1287	24
As Kh Qc Js 2d	1287	24
As Kh Qc Jd 2s	1287	24
Ah Ks Qs Jc 2d	1287	24
Ah Ks Qc Js 2d	1287	24
Ah Ks Qc Jd 2s	1287	24
Ah Kc Qs Js 2d	1287	24
Ah Kc Qs Jd 2s	1287	24
Ah Kc Qd Js 2s	1287	24

I think what he's asking for is the probability of each type of hold in the game he described...i.e the probability of being dealt a hand where you'll hold say Q, J off suit in 9/6 JoB.

Quote: Ibeatyouraces

Quote: ThatDonGuy

If I am reading a request right, here is a table of all possible hands:

Click Here

"Ranks" is how many different sets of ranks exist for that hand. For example, for the first one, the four of a kind can be any of the 13 ranks, and the single can be any of the 12 remaining ranks; 13 x 12 = 156.

"Suits" is how many different sets of suits exist for each set of ranks. For example, for the second one (the full house where both suits in the pair are also in the three), there can be four sets of three suits for the three of a kind, and for each set, three pairs of suits for the pair; 4 x 3 = 12.

Hand	Ranks	Suits
As Ah Ac Ad Ks	156	4
As Ah Ac Ks Kh	156	12
As Ah Ac Ks Kd	156	12
As Ah Ac Ks Qs	858	12
As Ah Ac Kd Qd	858	4
As Ah Ac Ks Qh	858	24
As Ah Ac Kd Qs	858	12
As Ah Ac Ks Qd	858	12
As Ah Ks Kh Qs	858	12
As Ah Ks Kh Qc	858	12
As Ah Ks Kc Qs	858	24
As Ah Ks Kc Qc	858	24
As Ah Ks Kc Qh	858	24
As Ah Ks Kc Qd	858	24
As Ah Kc Kd Qs	858	12
As Ah Kc Kd Qc	858	12
As Ah Ks Qs Js	2860	12
As Ah Kc Qc Jc	2860	12
As Ah Ks Qs Jh	2860	12
As Ah Ks Qh Js	2860	12
As Ah Kh Qs Js	2860	12
As Ah Ks Qs Jc	2860	24
As Ah Ks Qc Js	2860	24
As Ah Kc Qs Js	2860	24
As Ah Kc Qc Js	2860	24
As Ah Kc Qs Jc	2860	24
As Ah Ks Qc Jc	2860	24
As Ah Kc Qc Jd	2860	12
As Ah Kc Qd Jc	2860	12
As Ah Kc Qd Jd	2860	12
As Ah Ks Qh Jc	2860	24
As Ah Ks Qc Jh	2860	24
As Ah Kc Qs Jh	2860	24
As Ah Ks Qc Jd	2860	24
As Ah Kc Qs Jd	2860	24
As Ah Kc Qd Js	2860	24
As Ks Qs Js 2s	1287	4
As Ks Qs Js 2h	1287	12
As Ks Qs Jh 2s	1287	12
As Ks Qh Js 2s	1287	12
As Kh Qs Js 2s	1287	12
Ah Ks Qs Js 2s	1287	12
As Ks Qs Jh 2h	1287	12
As Ks Qh Js 2h	1287	12
As Ks Qh Jh 2s	1287	12
As Kh Qs Js 2h	1287	12
As Kh Qs Jh 2s	1287	12
As Kh Qh Js 2s	1287	12
Ah Ks Qs Js 2h	1287	12
Ah Ks Qs Jh 2s	1287	12
Ah Ks Qh Js 2s	1287	12
Ah Kh Qs Js 2s	1287	12
As Ks Qs Jh 2c	1287	24
As Ks Qh Js 2c	1287	24
As Ks Qh Jc 2s	1287	24
As Kh Qs Js 2c	1287	24
As Kh Qs Jc 2s	1287	24
As Kh Qc Js 2s	1287	24
Ah Ks Qs Js 2c	1287	24
Ah Ks Qs Jc 2s	1287	24
Ah Ks Qc Js 2s	1287	24
Ah Kc Qs Js 2s	1287	24
As Ks Qh Jh 2c	1287	24
As Ks Qh Jc 2h	1287	24
As Ks Qc Jh 2h	1287	24
As Kc Qs Jh 2h	1287	24
Ac Ks Qs Jh 2h	1287	24
As Kh Qs Jh 2c	1287	24
As Kh Qs Jc 2h	1287	24
As Kh Qc Js 2h	1287	24
As Kc Qh Js 2h	1287	24
Ac Ks Qh Js 2h	1287	24
As Kh Qh Js 2c	1287	24
As Kh Qh Jc 2s	1287	24
As Kh Qc Jh 2s	1287	24
As Kc Qh Jh 2s	1287	24
Ac Ks Qh Jh 2s	1287	24
As Ks Qh Jc 2d	1287	24
As Kh Qs Jc 2d	1287	24
As Kh Qc Js 2d	1287	24
As Kh Qc Jd 2s	1287	24
Ah Ks Qs Jc 2d	1287	24
Ah Ks Qc Js 2d	1287	24
Ah Ks Qc Jd 2s	1287	24
Ah Kc Qs Js 2d	1287	24
Ah Kc Qs Jd 2s	1287	24
Ah Kc Qd Js 2s	1287	24

I think what he's asking for is the probability of each type of hold in the game he described...i.e the probability of being dealt a hand where you'll hold say Q, J off suit in 9/6 JoB.

To add to this, not only am I looking for the probability of being dealt QJo in the game I outline above, but I'm looking for that probability net of all the hands above it on the strategy chart. I appreciate the try though Don.

Quote: Ibeatyouraces

I think what he's asking for is the probability of each type of hold in the game he described...i.e the probability of being dealt a hand where you'll hold say Q, J off suit in 9/6 JoB.

In that case, there is no easy way to do this - you pretty much have to use "brute force."

In other words, take each of the 134,459 "unique hands" (hands that are the same except for the specific suits - for example, a royal flush in spades is the same as one in hearts, and Ks Kh Kc Qs Qh (a full house where both cards in the pair have suits that are also in the three) is the same as Ks Kh Kd Qh Qd, but not Ks Kh Kd Qs Qc (as one of the queens is in a suit that is not also one of the kings)), then go down the strategy list in order until you find the hand ("Royal Flush? No...Straight Flush? No...Four of a Kind? No...Full House? Yes"), and add the total number of "actual" hands of that type to that strategy line (in this case, there are 12 different "versions" of Ks Kh Kc Qs Qh).

Quote: ThatDonGuy

In that case, there is no easy way to do this - you pretty much have to use "brute force."

In other words, take each of the 134,459 "unique hands" (hands that are the same except for the specific suits - for example, a royal flush in spades is the same as one in hearts, and Ks Kh Kc Qs Qh (a full house where both cards in the pair have suits that are also in the three) is the same as Ks Kh Kd Qh Qd, but not Ks Kh Kd Qs Qc (as one of the queens is in a suit that is not also one of the kings)), then go down the strategy list in order until you find the hand ("Royal Flush? No...Straight Flush? No...Four of a Kind? No...Full House? Yes"), and add the total number of "actual" hands of that type to that strategy line (in this case, there are 12 different "versions" of Ks Kh Kc Qs Qh).

Yeah, after spending quite a few hours on this myself, I definitely know it's not easy. ;) Was hoping there'd be some software out there that I could pick-up that would do the trick. Or, failing that, that a coder looking to make a few bucks on the side would see this, and we could work something out. Or, I suppose, I was also holding out hoping that someone substantially smarter than me would see this, and know a quick way to do it, and just whip it up real quick, lol.

Maybe the Wiz will add that feature to his strategy calculator since he likely has the values on hand.

I could answer the question but it would be time consuming. It would help if I knew of a reason this information would be helpful to anybody.

Quote: Wizard

I could answer the question but it would be time consuming. It would help if I knew of a reason this information would be helpful to anybody.

Well, I'm certainly happy to know that I'm not a moron for not being able to easily whip this up.

The reason I'm looking for this information is that there are progressive pools in the game, and I'm looking to figure out at what threshold it becomes +EV to play. This game is in just about every bar in my state (SD), and when I'm out and about, I like to check out the pools to see if there's anything worth playing, but I really don't know where it becomes +EV. I'm pretty sure once I have the data for the pre-draw hands, I can work out the thresholds for the pool amounts from there.

I know this isn't some super-sweet AP opportunity, just has the potential for a few Sklansky bucks here and there, so would be happy to share any and all information on this.

Okay, I’ll say it. You can compute the EV of games using the Wiz’s Video Poker Analyzer:
https://wizardofodds.com/games/video-poker/analyzer/

So if you see an interesting progressive pay table, just insert the current payout values and the analyzer will give you the EV.

Or you could iterate with the progressive outcome payouts to find breakeven values.

I’m not sure how you would be able to do this given the information you were seeking since optimal play varies as the progressive amounts change, so the probabilities you sought would not be static.

Quote: GaryJKoehler

Okay, I’ll say it. You can compute the EV of games using the Wiz’s Video Poker Analyzer:
https://wizardofodds.com/games/video-poker/analyzer/

So if you see an interesting progressive pay table, just insert the current payout values and the analyzer will give you the EV.

Or you could iterate with the progressive outcome payouts to find breakeven values.

I’m not sure how you would be able to do this given the information you were seeking since optimal play varies as the progressive amounts change, so the probabilities you sought would not be static.

This is what I tend to do with progressive games with the strategy calculator. The progressives would need to be huge in this game though, or there needs to be some type of other gimmick. With the payouts he gave, the payback is barely 85%.

Quote: tringlomane

This is what I tend to do with progressive games with the strategy calculator. The progressives would need to be huge in this game though, or there needs to be some type of other gimmick. With the payouts he gave, the payback is barely 85%.

The progressive on the machine has to do with when you have a 4 or a 9 in a losing hand. So, if I had the probabilities for the various starting hands, I can do the math from there and figure out where the +EV points are. Unfortunately, Wizard's analyzer doesn't have that information.

Quote: jek187

The progressive on the machine has to do with when you have a 4 or a 9 in a losing hand. So, if I had the probabilities for the various starting hands, I can do the math from there and figure out where the +EV points are. Unfortunately, Wizard's analyzer doesn't have that information.

Sorry, I completely missed that and recall prior correspondence on it. Not knowing the probability distribution they use to set the 4 and 9 jackpot points was the big unknown. I think you were going to assume a uniform distribution.

Quote: GaryJKoehler

Sorry, I completely missed that and recall prior correspondence on it. Not knowing the probability distribution they use to set the 4 and 9 jackpot points was the big unknown. I thi-nk you were going to assume a uniform distribution.

Yeah, how the jackpots were distributed is an unknown, but obviously not to that part yet, lol. I figure if I compute thresholds using a uniform distribution, and just play when they're above that, that even if the thresholds end up being a bit too low, I'll only be slightly -EV in those cases, and still +EV when the pools substantially exceed the threshold, that at worst I'll be somewhere near EV0, and can gather some data on when they payout, all the while impressing ladies at the bar that are turned on by this sort of analysis.

Quote: jek187

all the while impressing ladies at the bar that are turned on by this sort of analysis.

That was a joke right???

Then again, I'm not sure what kind of ladies you are trying to impress.

Quote: AxelWolf

That was a joke right???

Then again, I'm not sure what kind of ladies you are trying to impress.

I figured stuff like this would lead to me acquiring an army of groupies. Amazed my PM inbox isn't already overflowing with women throwing themselves at me. They're probably waiting for the prodigy that comes along and fills my request. That's gotta be it.