On the first hand dealt, how many ways are there for arranging 5 of 52 possible playing cards? How many of these possible arrangements are likely to produce winning final hands? If we were to consider all of the possible arangements, frequency of occurrence, and various possible winning outcomes for each hand, could we then make an accurate prediction of the chances of winning the hand? If so, what would be the Expected Return of any of these card combinations? If we knew this value, couldn't we then rank these hands in some logical order so that we would have a firm mechanism for choosing the cards to hold that have the highest probable return? The answers to these questions and a detailed understanding of the strategy behind successful Video Poker play is found with a quick look at an example and some simple equations related to Probability Theory.

Playing video poker with a 52 card deck will produce one of 311,875,200 possible card combinations on the initial hand dealt. Of this incredible number hands, there are 2,598,960 unique hand combinations if you disregard the order of the cards in the hand(which of course, doesn't matter for computing winning hands). Of these unique hands, we can make classifications or groupings of specific types of playable hands that have a certain probability or Expected Return given the combined odds of each of the possible winning outcomes for that hand. There are 36 such hands. Rather than printing out each of the 2,598,960 unique hands and then trying to decide on some arbitrary form of classification, each of the 36 classes can be derived using an efficient statistical structure.

After grouping similar classes we can accurately determine an Expected Return for each class using some simple equations from Probability Theory and an understanding of all possible ways to complete the hand on the final deal so that a win is incurred. For instance, let's say, on the first hand, you were dealt a 4-Card Outside Straight with 2 high cards such as the hand pictured below. Let's walk through the steps in calculating the Expected Return for this hand.

After the initial hand is dealt, there are 47 cards left in the deck. Although, in this case, we are trying to complete a straight (payoff of 4 coins per 1 coin bet), we also might get a pair of Jacks for a High Pair or a pair of Queens. So to calculate the Expected Return, we must take into account three factors: all possible winning outcomes, the number of coins per outcome, and the statistically proven odds of producing the winning outcome. Here we use the Classical Definition of Probability:

If a trial can result in n equally likely events and if m of these events are favorable to the occurrence of an event E, then the probability P(E) of the event E occurring is equal to the number of favorable events divided by the number of possible events.

With this theory, we can calculate precisely the chances of being dealt a final winning hand. Given the number of possible card combinations in the final deal, and the number of winning cards left in the deck, we can compute our probability of winning that hand type. 

Because it is true that the probability of the sum of mutually exclusive events is equal to the sum of the probabilities of these events(P(E1+E2)), the Expected Return can be calculated by combining the results of multiplying the number of coins paid for each winning hand possibility by the corresponding chance of completing that hand. In this case, holding the four cards for the final hand provides the following possibilities for a return:


8 chances to complete a straight with an 8 or a King E1 = 8/47*(4)=.681

6 chances to complete a high pair with Jack or Queen E2 = 6/47*(1)=.128

So here, P(E1+E2) is equal to .809 or .81, the Expected Return or Value for this hand. It is the Expected Return that allows us to rank each of the different categories of initial hands. And it is this ranking that governs the Smart Play feature, the Simulation Runs, and should soon be governing your play the next time you venture into a casino.