Calculate RTP?!?!?

I'm making a new game and it basically is a 3 card stud game. One bet, three cards you vs. the dealer. In order to get paid on the bet the dealer must have a queen or higher. According to 3 card poker data a player will win 44.91% of the time with these rules. What's the formula knowing the winning percentage and all the payout probabilities?

Quote: richardsonnerd

I'm making a new game and it basically is a 3 card stud game. One bet, three cards you vs. the dealer. In order to get paid on the bet the dealer must have a queen or higher. According to 3 card poker data a player will win 44.91% of the time with these rules. What's the formula knowing the winning percentage and all the payout probabilities?
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Is this just a straight up Even Money game, or are there other rules?

I'm trying to calculate the pay tables to create the house edge then rtp. Right now I know that after hiring someone to generate our rtp, that if the pay table is as follows then our RTP is 89.57%. I'm trying to find the formula for that or the house edge so I can tweak the payouts.

Mini Royal: 30 to 1
Straight Flush: 15 to 1
Three of a Kind: 10 to 1
Straight: 3 to 1
Flush: 2 to 1
Pair: 1 to 1
High Card: 1 to 1

The dealer must qualify with a queen or better or it is a push. In order to get the payout the player must beat the dealer. I know that a dealer only qualifies 69.59% of the time. I have all the probabilities per hand and the returns. I just don't know the formula to calculate the house edge. My reverse engineering is failing.

Quote: richardsonnerd

I'm trying to calculate the pay tables to create the house edge then rtp. Right now I know that after hiring someone to generate our rtp, that if the pay table is as follows then our RTP is 89.57%. I'm trying to find the formula for that or the house edge so I can tweak the payouts.

Mini Royal: 30 to 1
Straight Flush: 15 to 1
Three of a Kind: 10 to 1
Straight: 3 to 1
Flush: 2 to 1
Pair: 1 to 1
High Card: 1 to 1

The dealer must qualify with a queen or better or it is a push. In order to get the payout the player must beat the dealer. I know that a dealer only qualifies 69.59% of the time. I have all the probabilities per hand and the returns. I just don't know the formula to calculate the house edge. My reverse engineering is failing.
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You could do this in excel with just the combinations of each hand.

Seems to me that the RTP is too low for this to be a game players will play. And the dealer qualifier missing 3/10 is brutal too. Too many pushes will slow the game too much from casino perspective I think.

What happens if J or better qualifies, and then you can tweak the pay table a bit to make the RTP closer to three card?

Yeah, the RTP is low, that's why I am trying to find the formula to tweak for something in the high 90's. This bet is actually a prop bet on a larger game but this was the easiest way to simplify it.

When you say do this in excel, I'm trying to set that up using the wizard of odds ante return table to guide me. However, I'm just confused on how to derive the formulas. ( i wish i could attach a screen shot) For instance let's narrow to a straight flush. The site says this:

Player wins with straight flush: combinations: 617,044 probability: .001515
Player has straight flush, dealer doesn't qualify: combinations: 266,196 probability: .000654
Player loses with straight flush: combinations: 968 probability: .000002
Player ties with straight flush: combinations: 144 probability: .0000005

If the odds of getting a straight flush are 0.002172, then those numbers added equal that number? How did the combinations get calculated? Probability?

If I'm looking at the chart correctly then I add up all of the wins, loses, ties, no qualifies in order to find my edge. Is that correct? BTW, I appreciate your help, greatly!

Ah ha! I think I figured it out, sorta. Any ideas how I find the missing combination numbers below? I have found a few online but have no idea the formulas on how to get them.

Player Win
Mini Royal:
Straight Flush:
Three of a Kind: 665,776
Straight: 8,975,484
Flush:
Pair:
High Card:

Player Win No Dealer Qualify
Mini Royal:
Straight Flush:
Three of a Kind: 288,960
Straight: 4,001,004
Flush:
Pair:
High Card:

Player Tie
Mini Royal: 12
Straight Flush: 132
Three of a Kind: 0
Straight: 18,288
Flush:
Pair:
High Card:

Player Loss
Mini Royal:
Straight Flush:
Three of a Kind: 3,312
Straight: 270,504
Flush:
Pair:
High Card:

I think you would need to brute force this if it’s not on Wizard of Odds three card page.

It's not on there in the numbers I need. I have my mini royals separate from the straight flush so that blows those numbers up. And the numbers on there only go to straight. Then it groups the flushes and below since the payout is for the ante and ante bonus. Was just curious if some awesome math nerd had these handy. Thanks!

Check out the tables here: 3 Card Poker Analysis I think all the numbers you want are on a table; specifically check out the Ante Bet Analysis Table.

To calculate yourself, you could use either combination theory to write out algebraic formulas, or you could write out all the hand combinations that are Q-high or higher.

I believe the Wizard uses a computer code that would treat this as a 6 card problem (3 cards each for dealer and player) and then automatically creates all possible combinations of six cards configured as (player hand+ dealer's hand) and accumulates the needed statistics. That's a lot of work but its how to do the calculations so that the answers are rigorously correct.

By the way, how is your game idea different than conventional Three Card Poker? They sound identical.

Quote: unJon

I think you would need to brute force this if it’s not on Wizard of Odds three card page.
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Here's what I get, although I am not entirely confident with the numbers:

Hand	Win	Tie	Lose	Not Qualified
Mini-Royal	36,244	12	0	37,440
Straight Flush	425,184	132	968	384,372
3 Of A Kind	500,656	0	3312	454,080
Straight	6,637,980	18,288	270,504	6,338,508
Flush	9,291,176	3288	1,260,596	9,637,644
Pair	24,210,864	11,232	11,873,520	32,883,840
Garbage	13,306,860	149,352	144,612,792	144,821,556

This is a RTP of 82.92%

That's close. I wonder if all the totals equals 407,170,400 combinations because the wizard of odds site says that a win on a straight flush has the combination of 617,044 which includes the mini royals. So wouldn't the sum of those two still equal 617,044? The tie and loss are correct. No qualify should equal 266,196 with the sum of the no qualify for the mini royal and the straight flush.

Quote: ThatDonGuy

Quote: unJon

I think you would need to brute force this if it’s not on Wizard of Odds three card page.
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Here's what I get, although I am not entirely confident with the numbers:

Hand	Win	Tie	Lose	Not Qualified
Mini-Royal	36,244	12	0	37,440
Straight Flush	425,184	132	968	384,372
3 Of A Kind	500,656	0	3312	454,080
Straight	6,637,980	18,288	270,504	6,338,508
Flush	9,291,176	3288	1,260,596	9,637,644
Pair	24,210,864	11,232	11,873,520	32,883,840
Garbage	13,306,860	149,352	144,612,792	144,821,556

This is a RTP of 82.92%
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You can check the table here: 3 Card Poker Analysis. I did a spot check on the "straight flush wins-dealer qualifies" combinations and thought I observed a small difference. just trying to be helpful.

Edit: same basic message as previous post.

Quote: gordonm888

Quote: ThatDonGuy

Quote: unJon

I think you would need to brute force this if it’s not on Wizard of Odds three card page.
link to original post

Here's what I get, although I am not entirely confident with the numbers:

Hand	Win	Tie	Lose	Not Qualified
Mini-Royal	36,244	12	0	37,440
Straight Flush	425,184	132	968	384,372
3 Of A Kind	500,656	0	3312	454,080
Straight	6,637,980	18,288	270,504	6,338,508
Flush	9,291,176	3288	1,260,596	9,637,644
Pair	24,210,864	11,232	11,873,520	32,883,840
Garbage	13,306,860	149,352	144,612,792	144,821,556

This is a RTP of 82.92%
link to original post

You can check the table here: 3 Card Poker Analysis. I did a spot check on the "straight flush wins-dealer qualifies" combinations and thought I observed a small difference. just trying to be helpful.
link to original post

I believe I did the same. You mentioned "You can check the table here: " but there was no link to click. Are you able to send again?

Quote: richardsonnerd

Quote: gordonm888

Quote: ThatDonGuy

Quote: unJon

I think you would need to brute force this if it’s not on Wizard of Odds three card page.
link to original post

Here's what I get, although I am not entirely confident with the numbers:

Hand	Win	Tie	Lose	Not Qualified
Mini-Royal	36,244	12	0	37,440
Straight Flush	425,184	132	968	384,372
3 Of A Kind	500,656	0	3312	454,080
Straight	6,637,980	18,288	270,504	6,338,508
Flush	9,291,176	3288	1,260,596	9,637,644
Pair	24,210,864	11,232	11,873,520	32,883,840
Garbage	13,306,860	149,352	144,612,792	144,821,556

This is a RTP of 82.92%
link to original post

You can check the table here: 3 Card Poker Analysis. I did a spot check on the "straight flush wins-dealer qualifies" combinations and thought I observed a small difference. just trying to be helpful.
link to original post

I believe I did the same. You mentioned "You can check the table here: " but there was no link to click. Are you able to send again?
link to original post

Link was in his post.

https://wizardofodds.com/games/three-card-poker/

Are you able to share this chart as maybe a starting point for me? Is it an excel file? I think you are close here. I've been on this for days! I would love to get this resolved.

Quote: gordonm888

You can check the table here: 3 Card Poker Analysis. I did a spot check on the "straight flush wins-dealer qualifies" combinations and thought I observed a small difference. just trying to be helpful.

Edit: same basic message as previous post.
link to original post

Good catch - thanks. I was counting ace-high hands as "ace-low", so it was overcounting the number of non-qualifying hands.
Here are revised numbers, that appear to match what the Wizard has, at least with straight flushes (when you include Royals):

Hand	Win	Tie	Loss	Not Qualified
Royal	46,804	12	0	26,880
Straight Flush	570,240	132	968	239,316
3 Of A Kind	665,776	0	3312	288,960
Straight	8,975,484	18,288	270,504	4,001,004
Flush	12,793,544	3288	1,260,596	6,135,276
Pair	36,168,624	11,232	11,873,520	20,926,080
Garbage	38,197,044	246,888	172,254,864	92,191,764

This has an RTP of 89.64%

Quote: ThatDonGuy

Quote: gordonm888

You can check the table here: 3 Card Poker Analysis. I did a spot check on the "straight flush wins-dealer qualifies" combinations and thought I observed a small difference. just trying to be helpful.

Edit: same basic message as previous post.
link to original post

Good catch - thanks. I was counting ace-high hands as "ace-low", so it was overcounting the number of non-qualifying hands.
Here are revised numbers, that appear to match what the Wizard has, at least with straight flushes (when you include Royals):

Hand	Win	Tie	Loss	Not Qualified
Royal	46,804	12	0	26,880
Straight Flush	570,240	132	968	239,316
3 Of A Kind	665,776	0	3312	288,960
Straight	8,975,484	18,288	270,504	4,001,004
Flush	12,793,544	3288	1,260,596	6,135,276
Pair	36,168,624	11,232	11,873,520	20,926,080
Garbage	38,197,044	246,888	172,254,864	92,191,764

This has an RTP of 89.64%
link to original post

Holy crap! You rock! I appreciate you so much! Thanks to everyone else for your help as well!

Quote: ThatDonGuy

Quote: gordonm888

You can check the table here: 3 Card Poker Analysis. I did a spot check on the "straight flush wins-dealer qualifies" combinations and thought I observed a small difference. just trying to be helpful.

Edit: same basic message as previous post.
link to original post

Good catch - thanks. I was counting ace-high hands as "ace-low", so it was overcounting the number of non-qualifying hands.
Here are revised numbers, that appear to match what the Wizard has, at least with straight flushes (when you include Royals):

Hand	Win	Tie	Loss	Not Qualified
Royal	46,804	12	0	26,880
Straight Flush	570,240	132	968	239,316
3 Of A Kind	665,776	0	3312	288,960
Straight	8,975,484	18,288	270,504	4,001,004
Flush	12,793,544	3288	1,260,596	6,135,276
Pair	36,168,624	11,232	11,873,520	20,926,080
Garbage	38,197,044	246,888	172,254,864	92,191,764

This has an RTP of 89.64%
link to original post

This is all perfect. I added it to my existing numbers and it all works out. Question from these numbers, what's the formula to calculate RTP? How did you arrive at 89.64%?

Quote: richardsonnerd

Quote: ThatDonGuy

Quote: gordonm888

You can check the table here: 3 Card Poker Analysis. I did a spot check on the "straight flush wins-dealer qualifies" combinations and thought I observed a small difference. just trying to be helpful.

Edit: same basic message as previous post.
link to original post

Good catch - thanks. I was counting ace-high hands as "ace-low", so it was overcounting the number of non-qualifying hands.
Here are revised numbers, that appear to match what the Wizard has, at least with straight flushes (when you include Royals):

Hand	Win	Tie	Loss	Not Qualified
Royal	46,804	12	0	26,880
Straight Flush	570,240	132	968	239,316
3 Of A Kind	665,776	0	3312	288,960
Straight	8,975,484	18,288	270,504	4,001,004
Flush	12,793,544	3288	1,260,596	6,135,276
Pair	36,168,624	11,232	11,873,520	20,926,080
Garbage	38,197,044	246,888	172,254,864	92,191,764

This has an RTP of 89.64%
link to original post

This is all perfect. I added it to my existing numbers and it all works out. Question from these numbers, what's the formula to calculate RTP? How did you arrive at 89.64%?
link to original post

Multiply all the probabilities by the outcomes (including 0 outcome for pushes and -1 outcome for a loss) then add them up.

Quote: ThatDonGuy

Quote: gordonm888

You can check the table here: 3 Card Poker Analysis. I did a spot check on the "straight flush wins-dealer qualifies" combinations and thought I observed a small difference. just trying to be helpful.

Edit: same basic message as previous post.
link to original post

Good catch - thanks. I was counting ace-high hands as "ace-low", so it was overcounting the number of non-qualifying hands.
Here are revised numbers, that appear to match what the Wizard has, at least with straight flushes (when you include Royals):

Hand	Win	Tie	Loss	Not Qualified
Royal	46,804	12	0	26,880
Straight Flush	570,240	132	968	239,316
3 Of A Kind	665,776	0	3312	288,960
Straight	8,975,484	18,288	270,504	4,001,004
Flush	12,793,544	3288	1,260,596	6,135,276
Pair	36,168,624	11,232	11,873,520	20,926,080
Garbage	38,197,044	246,888	172,254,864	92,191,764

This has an RTP of 89.64%
link to original post

Thank you so much for all of your help with these three card numbers. Since you are pretty much the Jesus of numbers in my book, I'm thinking of converting to a 5 card game too. Can you whip out the same exact chart for the 5 card combinations? Royal Flush, Straight Flush, Four of a Kind, Full House, Flush, Straight, Three of a Kind, Two Pairs, One Pair, High Card. That would be awesome!

Those are single player five card poker probabilities. Straight flushes can be further divided as 4 combinations for Royal Flushes and 36 combinations for non-royal straight flushes. You would need to decide what is a qualifying hand for the dealer. Also, be aware that 5 card poker games against the dealer already exist. and some have patent protection.

Also, I would advise you to work harder at searching the internet for basic results like this and learning how to do calculations yourself. You might start by looking up games on the Wizard of Odds site, but the internet has many good sites with information about things like poker probabilities -even Wikipedia.

Many of us like to be helpful, but its a bit unreasonable to suppose that every time that you want to explore a game concept that someone on this site is going to be willing to do analysis for free.

Quote: gordonm888

Those are single player five card poker probabilities. Straight flushes can be further divided as 4 combinations for Royal Flushes and 36 combinations for non-royal straight flushes. You would need to decide what is a qualifying hand for the dealer. Also, be aware that 5 card poker games against the dealer already exist. and some have patent protection.

Also, I would advise you to work harder at searching the internet for basic results like this and learning how to do calculations yourself. You might start by looking up games on the Wizard of Odds site, but the internet has many good sites with information about things like poker probabilities -even Wikipedia.

Many of us like to be helpful, but its a bit unreasonable to suppose that every time that you want to explore a game concept that someone on this site is going to be willing to do analysis for free.
link to original post

I understand, thanks for the feedback. I do appreciate the help. Those numbers you displayed are easy to find, I have searched. The person that responded to the possible winning three card combinations had the ability to quickly provide information that honestly I couldn't find after days of searching. The five card numbers appear to be just as difficult broken down into Win, Lose, No Qualify, Tie, like he gave me, that's exactly what I need. If you know of a site that has this chart, I would love that link!

As for the hand that qualifies, it appears that Caribbean Stud uses A,K. A pair or higher could also work for qualifying hand.

As for patented games, this game is a spin off of our 3 Card War game that is actually patented by us, Neesan Labs, we have a 5 Card War and a War Poker game all internationally patented as well. The poker bet is a prop bet in addition to the basic game ante/war bet where other prop bets are included. We just ran out of money in the building process and can't afford a math guy as we look to secure investors for GLI-19 certification. I was actually going to offer a small payment to the person who helped with the 3 Card Poker combinations but I assumed that was against the forum rules. Instead I was going to offer him a credit in our games but ran out of available posts yesterday since my account is new and limited. If I can offer him a small payment through here, I'd be glad to. Or you if you can duplicate the chart he gave me for 3 card stud, I'd be open to offering a payment or game credit to you as well.

I, for one, am glad to hear the offer of compensation.
Best of luck with the certification process.

Quote: richardsonnerd

Thank you so much for all of your help with these three card numbers. Since you are pretty much the Jesus of numbers in my book, I'm thinking of converting to a 5 card game too. Can you whip out the same exact chart for the 5 card combinations? Royal Flush, Straight Flush, Four of a Kind, Full House, Flush, Straight, Three of a Kind, Two Pairs, One Pair, High Card. That would be awesome!
link to original post

That's a little more work that I can find time to do at the moment.

Quote: ThatDonGuy

Quote: richardsonnerd

Thank you so much for all of your help with these three card numbers. Since you are pretty much the Jesus of numbers in my book, I'm thinking of converting to a 5 card game too. Can you whip out the same exact chart for the 5 card combinations? Royal Flush, Straight Flush, Four of a Kind, Full House, Flush, Straight, Three of a Kind, Two Pairs, One Pair, High Card. That would be awesome!
link to original post

That's a little more work that I can find time to do at the moment.
link to original post

Let me know if it is a possibility in the next day or so. We are willing to compensate even though it wouldn't be as much as we'd like. I appreciate all you have helped me with thusfar. Thank you!

Quote: richardsonnerd

Quote: ThatDonGuy

Quote: richardsonnerd

Thank you so much for all of your help with these three card numbers. Since you are pretty much the Jesus of numbers in my book, I'm thinking of converting to a 5 card game too. Can you whip out the same exact chart for the 5 card combinations? Royal Flush, Straight Flush, Four of a Kind, Full House, Flush, Straight, Three of a Kind, Two Pairs, One Pair, High Card. That would be awesome!
link to original post

That's a little more work that I can find time to do at the moment.
link to original post

Let me know if it is a possibility in the next day or so. We are willing to compensate even though it wouldn't be as much as we'd like. I appreciate all you have helped me with thusfar. Thank you!
link to original post

I suggest that you take any further for-fee discussions to the Private Messaging system.