What Does 'Expected Value' Really Mean? "All professional gamblers rate games according to the 'expected value.' This concept, in fact, is so important to online casino gamblers that, as with Yahweh of old, you rarely hear the words spoken-just the initials EY. No professional would ever place a bet on any game if he or she didn't think that the EV was positive." Arnold "The Bishop" Snyder, Card Player.
Although "Bishop" Snyder writes mostly about blackjack, the entire article from which the above quote is extracted should be read by all serious players of any game, but it is especially pertinent to us because video poker is one of the few games where the EV can be precisely determined and you can get an advantage over the game.
But just what is expected value?
Simplest Casino Gambling Terms
To put it in as simple terms as possible, the EV of any chance event is the average of all possible outcomes. But now we must define the word "average" which to a statistician may have any of several meanings depending on the context and the question to be investigated. In this case it is a weighted mean, computed by multiplying the probability of each possible outcome by its value (payoff per unit bet) and summing the products. Now let's see how to determine the EV of one video poker play.
For any dealt five-card hand, there are thirty-two possible ways to play it. You can hold all five cards (That's one way.); you can hold four cards and draw one (Obviously there are five possible one card discards.); you can hold two cards and draw three (There are ten possible combinations for a two-card discard.); you can hold three cards and draw two (again ten possibilities); you can hold one card and draw four (There are five ways to hold just one card.); or you can discard all five cards. In any case, you have seen five of the 52 cards in the deck, so the draw must come from the remaining 47 unseen cards.
For each of these 32 ways of playing a hand, there may be many possible outcomes. Let's look at just a few of the possible ways of playing the following 9/6 Jacks or Better hand: K. Q. J 10. 4. Draw 1: Hold K-Q-J-l0, hoping for a straight. Any ace or nine will make a straight, which pays 4-for-l, and there are eight such cards in the deck. But that's not all. Any king, queen or jack will make a high pair, which pays I-for-I, and there are a total of nine such cards remaining in the deck.
How to Draw
Draw 1: Hold K-J-10-4 (all hearts), hoping for a flush. Any heart will make a flush which pays 6-for-l, and there are nine such cards remaining in the deck.
Draw 2: Hold K-J-l0 (all hearts), hoping for a royal flush. This is more complicated because you are drawing two cards instead of one. Since there are 1,081 possible outcomes instead of just 47, I'll skip part of the mathematics. Obviously there is only one possible way to male a royal (by drawing the ace and queen of hearts) which pays 800-for-l (assuming five-coin play and a 4,000-coin royal). There is also just one way to make a straight flush (by drawing the queen and nine of hearts) which pays 50-for-I. There are 34 ways to draw two hearts other than A-Q or Q-9 to make a flush which pays 6-for-l, 22 ways to draw an A-Q or Q-9 (not both hearts) to make a straight which pays 4-for-l, nine ways to draw a pair of kings, jacks or tens to make three of a kind which pays 3for-I, 27 ways to make two pair (e.g., by catching a king and a ten) which pays2-for-l, and 237 ways to make a high pair (by catching just one king or one jack or a pair of aces or queens) which returns your bet.
