# Fight Math With Math

Have you ever been dealt a hand that contains 3 of one suit and 2 of another and wondered why you don't get paid for this type of 'Full house'?  Imagine if video poker paid for this type of hand.  It is hard to say the exact probability of getting this type of hand after a draw.  I know that a little over 10% of hands are dealt this way, but a fair number of them are other winning hands or near winners.  If you had 3 diamonds to a Royal and 2 clubs would you go for the Royal or would you just take your suit Full House and be content.  You can't answer this until you know how much this hand pays.  If I said it paid 1, you might go for the Royal.  If I said it pays 4, you'd probably keep the suit Full House.

Of course to pay for such a hand, you'd have to lower the pays on many other hands and perhaps even eliminate paying for some.  Maybe you'd only be able to pay for Kings or better instead of Jacks.  Since I don't know the exact frequency of our new hand, I don't know exactly how much it is going to contribute to the overall payback and thus I don't know how much I need to lower or eliminate other paybacks.

Just for fun, let's pretend that we're going to pay 1 for this new hand and that at a payout of 1, it will occur about 20% of the time.  As luck would have it, a High Pair occurs about 21-22% of the time, so this is a near wash.  We can simply substitute our suit Full House for a High Pair and we would have invented a new video poker game with a slightly lower payback than regular video poker.  The casinos should love it!

Not so fast.  The problem is that if I were to put this new hand into a program that determines strategy and calculates payback, it is not going to find this trade off so simple.  Getting rid of High Pairs is going to change the expected value of virtually every hand.  Adding this new suit Full House is also going to change the expected value of many hands.  I don't even want to think what it will do to our strategy table.  Now, not only will we have to worry about High Cards, we're going to have to think about whether or not the hand has the potential to wind up as this winning hand.  Have a 4-Card Straight with 2 cards each from 2 suits - great.   If it has 3 suits, it can't wind up as a suit Full House winner and its expected value will be lower.  The strategy table will get 20% longer.

No, someone did not pitch this idea to me this past week.  I bring up this scenario to show how video poker is a series of moving parts.  There used to be a board game called Stay Alive (or Staying Alive?) where you put a marble on the board and then each Player could take turns moving a series of levers.  The goal was to get the opponents marbles to fall to the bottom while keeping yours on the board.  When you pulled (or pushed a lever), sometimes you could see the hole the marble would fall into.  Sometimes it would land one lever down and sometimes it would fall to the bottom.  Making changes to video poker is a lot like that game.  You can tweak a paytable by changing a single pay.  That change will ripple through the entire strategy table.  As the strategy changes, some hands will become more common, while others will become less.  Usually, the change is very small.  But a bunch of small changes can add up.

Maybe your changes are larger.  You don't just tweak a pay, but you essentially add hands to the paytable.  This sort of happens in some of the bonus games where having a particular kicker as a 5th card for a Four of a Kind pays more (i.e. 4 Aces with a 2-3-4).  Depending on the paytable, it is theoretically possible that you would hold Two Pair if you have Aces and 2's, but hold only the Pair of Aces if you have Aces and 10's and the 5th card is not a 2, 3 or 4.  As an analyst, I have to put the games together in a way that takes into account all these moving parts.  As a Player, you have to take into account the final product when determining the strategy.  In many respects, your job is easier because you only have to worry about the one scenario that actually exists, while I may have to go through 10 or 20 possibilities before a decision is made on which one(s) will be used.  It also helps that you simply need to use the results of my work and you don't have to do the work yourself.

My point is not that if I have to do all this work, so do you!  The point is that if all this math goes into building the product, then your best chance for winning is to use that very same math against the game.