A few months ago, I volunteered at my son's school at their book fair. During slow periods, the parents shoot the breeze while getting to know one another. My job is relatively unique, so it tends to get a lot of attention. This inevitably leads to me telling the story about how my dad got started analyzing casino games. It started one day when he walked into a casino and saw a video poker machine. This was the late 1980's (maybe early 1990's?). A short time later he walked into a different casino and saw a similar video poker machine. The two machines advertised very different paybacks. My dad didn't understand how two machines with identical paytables could have different paybacks.
Now, if these were slot machines, this would not be such a surprise. At the time, my father surmised that maybe this was what was going on with the video poker machines. But, he did some investigative work and found out that a 'slot machine' that used an electronic version of a real world object had to play as randomly as the real world item. In other words, if it used a deck of cards, it had to play just as if it you were dealing a REAL deck of cards. So, each card was random, but the outcome of the deal and draw were tied to probabilities that could be calculated with certainty.
Slot machines don't work the same way. You may see 20 symbols on a real, but this doesn't mean that each has a 5% probability of appearing in any given spin. If there are 4 reals on the slot machine, some of the apparent combination may NEVER appear. They can be programmed this way completely legally. In video poker, all 2,598,960 combination of 5-card deal will show up - each with the exact same probability as the next. How did we arrive at this number? This is the number of ways that you can deal 5 random cards from a 52 card deck. Mathematically, it is called '52 choose 5' and it is (52 x 51 x 50 x 49 x 49) / (5 x 4 x 3 x 2 x 1).
Now, from a Poker standpoint, each rank of hand will not occur with the same probability. It is much harder to be dealt a four of a kind than it is 5 distinct cards of different suits that also don't make up much of a Straight. But, the magic of video poker is that you get a DRAW as well as a DEAL. This Draw has a very important component to it. It is that the Player gets to decide how many cards and which cards he wants to discard and have replaced. So, unlike many table games, the Player doesn't add to his wager and/or decide to fold in the middle of the game. Instead he gets to replace some of his cards. But, strategy is strategy and it becomes a key part of the game.
Essentially, video poker is to slots what blackjack is to baccarat. In baccarat, you make a decision to bet on the Banker or Player hand. After that the decision to draw cards is not in anyone's hands. You're done after you make your wager. In blackjack, you make your initial wager and you get dealt 2 cards. Now you have to decide what to do. Will you hit, stick, double, split? You have different decisions in video poker but the concept is very similar. Like in blackjack, many hands require very little thought. You're dealt a Pair of Aces, along with a 2 - 6- 9? That's as obvious as what do with a Hard 19 vs a 6. Change that to a Pair of 4's and have the 2-6-9 be suited and the same suit as one of the 4's and now you have something to think about. Pair of 4's or 4-card Flush?
Like in blackjack, the decision of what to do is calculated mathematically. In blackjack, the math is a bit easier as there are only two choices (at most) in most cases. The decision to hit or stick is dictated by which provides the greater return for the Player. In the simplest terms, which strategy will cause the Player to win the hand more often? The decision is video poker is a bit more complex because the hand can have different payouts depending on the final rank. But, again, conceptually, it is the same. In video poker, we look at each possible deal for each potential draw situation. In the case of the 4-card Flush vs. the Low Pair, we look at the 47 possible draws for the 4-card Flush and add up the potential payout of all 47. We do the same for the 16,215 possible draws for the Low Pair. To be able to compare apples to apples, we divide the total by the number of possible draws for each. We then compare the result, which we call the expected value. The one with the higher expected value is the proper strategy.
In reality, we use a computer program to do this work and we look at each of the 32 possible draws that the Player can make. Most of the time at least 29 of these ways would never be considered, but it is hard (but not impossible) to tell a computer how to eliminate the obviously wrong choices. Given the speed of today's computers, it is easier to let them crank thru all the possibilities, just in case and provide a summary of the results. This is how we create the strategy for a particular paytable and figure out the payback associated with that paytable.
Somehow, my dad figured this all out about 25 years ago. He never really did get an answer for how two different machines with the same paytable could claim different paybacks. At least one of the two machines advertised payback was incorrect. We've always assumed that the people who created these first games created an intelligent, but less than perfect strategy that the thought people would play and didn't concern themselves with how far off they might be.