(Yes, I think you're thinking about combinations correctly. It "should" be obvious that the 4-of-a-kind numbers will change between games where some 4oK's are treated as premium hands compared to others, and change further if there are kickers involved (4Aw/2,3,4), and change again if there are wildcards.)

Variance is a statistics term, used to describe how widely the set varies from 1. The less formal term is "volatility".

The low-math pragmatic explanation: the higher the variance, the worse your day will be if you don't hit a big hand (like aces with a kicker or a royal flush).

Welcome to the forum.

Best of luck.

Quote:4mylvle…1). Combinations. When looking at game analyzers what is combinations? Is this possible combinations to achieve the stated hand? If so, why are the combinations different for all games utilizing a single 52 card deck? And, why do they change with payscale? Is the true amount of combinations an impossibility to write into code? Or, am I just thinking in the wrong way about the term combinations?…

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Yes, it is the possible combinations to achieve the stated hand, while taking into account the probabilities on the deal, holding the cards with the highest expected value for the given pay table, and the probabilities on the draw. Each unique possibility is included in the calculation.

Quote:Dieter…Variance is a statistics term, used to describe how widely the set varies from 1…

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Not exactly, but in most Video Poker, it is close to correct. Variance is used to describe how widely the set varies from the mean, or average, value. In any game in the house, this is merely the return stated as a decimal. In full pay Jacks, the mean would be 0.9954… While this is close to 1, and it probably wouldn’t alter the variance calculation too much to use a rounded figure as the mean in this case, I wouldn’t want to estimate the return this way!

Feel free to ask more questions. Many of us will try to help if we can!

Quote:camaplQuote:Dieter…Variance is a statistics term, used to describe how widely the set varies from 1…

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Not exactly, but in most Video Poker, it is close to correct. Variance is used to describe how widely the set varies from the mean, or average, value. In any game in the house, this is merely the return stated as a decimal. In full pay Jacks, the mean would be 0.9954… While this is close to 1, and it probably wouldn’t alter the variance calculation too much to use a rounded figure as the mean in this case, I wouldn’t want to estimate the return this way!

Feel free to ask more questions. Many of us will try to help if we can!

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I was today years old when it started to make more sense.

Mean as RTP definitely clicks.

Quote:4mylvleCombinations. When looking at game analyzers what is combinations?

It may depend on what is being analyzed. If you mean something like on the Wizard's page, where it lists a total of 19,933,230,517,200 "combinations," this is the 2,598,960 possible 5-card hands you can be dealt, multiplied by the 1,533,939 sets of 5 cards that can be the next 5 in the deck, divided by 2. "Why divided by 2?" Because all of the other numbers were multiples of 2, and he reduced all of them to make it easier to calculate. (In the "40-20-9-6-5 Net Entertainment Pay Table" results, the total number of combinations is "only" 1,661,102,543,100 because all of the other numbers were multiples of 24, so they could all be divided by 24.

In mathematics, a "combination" is a set of things where the order doesn't matter. This differs from a "permutation," where the order does matter. For example, when counting the number of 5-card hands you can be dealt, the order does not matter, so there are 2,598,960 ways. However, if there is a bonus for having a Royal Flush where the cards appear 10, Jack, Queen, King, Ace from left to right, then order does matter, and there are 2,598,960 x 120 = 311,875,200 different possible deals. There are 2,598,960 "combinations" and 311,875,200 "permutations" of 5 cards selected from 52.

Quote:4mylvleI assume that there are a finite number of dealt hand combinations and from those finite number of combinations there are a finite number of ways to make a royal flush. If this is the case, wouldn't the combinations for a royal flush be identical in any 52 card game? Why are the combinations for 9/6 JOB 41,126,022 but for 8/5 the combinations are 41,352,073? Same game, same deck of 52 but 226,051 more ways to get to the royal. How is this possible?

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There is almost certainly a difference in hold strategy where part of a royal flush is discarded in favor of a lesser hand, precluding a royal flush in one of the games.

Quote:4mylvleI feel like I'm not asking the right questions or my questions are not understood. I am asking about the very base of the game. Are the graphs in the analyzers based on all possible outcomes of a 5 card draw poker game with a 52 card deck or are they based on the possible outcomes using the ideal strategies of each different game? Maybe this is where I am getting confused. So I guess my question could also be simply put this way as I think through this, regardless of game or strategy, what is the total possible combination of holds to achieve a royal flush? I know that the initial deal is a total of 2,598,960 and from there, I assume there are a finite number of holds using every possible what if scenario to achieve a royal. Is that number incalculable? The way I'm defining combinations, that would be the number I expect to see in the combinations row of the graph. If the graphs are based on strategy rather than total possible outcomes then that's where I have been confused. Once I can work past this topic I have many more questions. Thanks!

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I'm not sure which graphs you're refering to.

I'm not sure which analysis software you're using.

That said, there are times when you're dealt 0 cards of a royal flush, and you hold at least 1 of them.

You cannot make a royal if you hold trip 3's.

There are times when you're dealt two Aces, and you generally hold the pair.

You cannot make a royal holding a pair of Aces.

I expect you're seeing numbers based on the optimal hold strategy for the paytable for each of the roughly 311.8 million (52*51*50*49*48) initial hands.

And yes, the numbers are calculable.

The light bulb is a little less dim today. Thanks for putting this in terms I can understand. I have been using simulators to practice strategies and it works but trying to achieve a fundamental understanding of the game is tough for a right brained person.Quote:camaplYes, the combinations are calculated using proper strategy. This is why they are different for each game and pay table, just as they are different when looking at perfect vs. basic strategy for the same game and pay table.

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Quote:4mylvleThe light bulb is a little less dim today. Thanks for putting this in terms I can understand. I have been using simulators to practice strategies and it works but trying to achieve a fundamental understanding of the game is tough for a right brained person.Quote:camaplYes, the combinations are calculated using proper strategy. This is why they are different for each game and pay table, just as they are different when looking at perfect vs. basic strategy for the same game and pay table.

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Everyone has already answered the questions greatly but I do feel like I can give a quick example on why 8/5 JoB has more combinations of royals assuming proper strategy, as already said, the combinations are calculated using proper strategy, I think it would help slightly to explain the (very slight) difference between 8/5 JoB and 9/6 JoB strategy.

In 9/6 JoB, there is an exception where a 4 to a flush actually beats 3 to a royal, I am not going to explain the exception as the wizard has it on his page.

For 8/5 JoB, because the flush only pays 5 instead of 6, the proper strategy actually calls for holding 3 to a royal more than holding 4 to a flush, therefore, there are more combinations of royals in 9/6 than 8/5 since you are very slightly not holding as much 4 to a flush.

Quote:4mylvleSo my next question is about variance, if someone can put it into terms I can understand. I get that variance is tied to how much bankroll is required to outlast the "lows" in a game. What I don't understand is where does it come from? Is it, 1). simply designed into the game software? Is it harder to achieve quad aces in a bonus game compared to a game that pays all quads equally? If the game is truly random wouldn't this be considered, for lack of a more intelligent term, not random? I am operating under the assumption that VP games are as close an approximation to playing with a real deck and a real shuffle as possible. With the odds as long as they are already, it would seem that the house edge would be much higher than it is with manipulated odds in favor of the house. Is variance simply the value of the engineered manipulation of the true odds of hitting any one hand? And if so, how are these numbers calculated? Is the manufacturer of the game required to make this information available? OR is it 2). simply shorting the low end of the payscale and the modified strategy that is responsible for variance? Things like throwing out aces full to go for the quad in a game like DDB or TB+???? OR is it 3). Neither of these things? Thanks!

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Wow, that’s a lot! But that’s ok, because it gives us an idea of what ideas you’re working with. Unfortunately, there are a lot of misconceptions out there. Not to worry, I have a pretty good handle on the math and I’ll admit that I fell for some of them before I found these sites.

The variance is calculated using the same figures used to calculate the return (or average). This means that like the return, the variance depends on the strategy as well. Obviously, the formulas for the return and variance are different, but they both utilize the same pay table and probabilities of occurrence of each hand based on strategy.

I don’t think of either the return or variance being “designed” into the programming. While the game makers do decide on which VP types and pay tables to offer the casinos, the return and variance are simply a function of (think “tied to”) the hand types and their payouts. It’s simply a one to one relationship, much as two plus two (always) equals four. The difference here is that the calculation is much more complex, so we’ll have rounding errors and subtle differences in strategies based on how accurate the strategy needs to be.

The different odds that come out of different games for the same hand (such as quad A’s) also has nothing to do with programming. If one plays both Double Double Bonus and Bonus Poker the same way (using the same strategy), the odds of quad A’s (with and without kicker) would be identical, as the games are programmed by the game makers to deal the game fairly. The difference in odds for all games and pay tables comes directly from the fact that each has a (potentially different) optimal strategy. The different payouts for each hand type changes the values of each set of cards that might be held. Optimal strategy has us hold or discard cards so as to give us the highest return (or average value of the final payout) based on which hands are possible for the cards held, if any. Your example in 2) regarding A’s full vs. 3 A’s is an excellent example of why the overall odds vary between the games. Clearly, you can see that keeping the full house vs. throwing the pair is going to change the overall odds of trips, full houses, and quads. Now add in all of the other idiosyncrasies between the different strategies, and you’ll see exactly what creates the different odds of each hand, and the return and variance for each game/pay table.