Win Frequency vs. Expected Value

            As far as I know, my father, Lenny Frome, pretty much coined the term Expected Value (or EV for short) where casino games are concerned.  He didn't create the concept of it, nor was he the first person who realized its importance in strategy.  But, he did give it a name and attempted to explain WHY it is so important to the masses. 

            Blackjack was one of the first casino table games with a true strategy.  We could argue to what extent that Craps or Roulette has a strategy, but I think we'd agree that if there is a strategy there it is a betting strategy, not a hand by hand type of strategy.  Every hand has its hit/stick/double/split/surrender strategy.  This strategy is impacted by how many cards the Player has (can't split or double if you have more than two cards), the Dealer upcard and in some cases the number of decks used in the game.  All of this strategy revolves around the same thing - the expected value of each choice.  Because of the betting structure and payouts of blackjack, there is a very strong correlation between win frequency and expected value.  This begins to diverge when you split and double.

            When you split, you create multiple hands which changes the equation a bit.  When you double, the equation completely changes.  By its very definition, you cannot increase win frequency when you double.  It can only stay the same or be reduced.  So, why would you do it?

            Because you get to double your wager, which means your expected value, as a percent of the original wager increases even as your win frequency might decrease.  The win frequency is meaningless.  Would you rather win 1 in 100 and win $10,000 or win 100 out of 100 and win $100 (in total)?  Not much to think about.

            Many of us are conditioned to thinking in terms of win frequency because that's what we see when we watch a traditional poker tournament on television.  As the cards are dealt, the screen shows us the probability of each player winning.  In a poker room, win frequency of a single hand is all that matters within that hand.  Whoever wins, wins the pot and there is no way to predict how big the pot will be.

            A game like video poker is, however, nothing like its table counterpart.  You're not playing against anyone.  You're playing against a paytable.  'Winning' with a Full House pays vastly more than winning with Two Pair.  While there is still some correlation between win frequency and expected value, they diverge far more often than in blackjack.  Very little sums this up more than the Low Pair.

            The Low Pair will win 28.7% of the time.  I'm pretty sure this is the highest win frequency of any hand that is not a guaranteed winner.  Frequently a hand will be both a Low Pair and either a 4-Card Straight or a 4-Card Flush.  If the hand is a 4-Card Flush as well, we discard the Low Pair and hold the 4-Card Flush, which has a win frequency of only 19.1% if there are no High Cards in the Flush.  Clearly, win frequency is not the driving force behind our strategy.  The expected value of the Low Pair is only 0.82 while the expected value of the 4-Card Flush is 1.15.

            Then there are the hands that are guaranteed winners that we actually discard!  Dealt a High Pair?  Throw it out if you have a 4-Card Straight Flush (Inside or not).   Dealt a Straight or a Flush?  Discard it if you have a 4-Card Royal.  Obviously, these are not common hands, but we are talking about discard a hand with a win frequency of 100% to keep a hand with a win frequency in the 25-30% range. 

            We do this because Expected Value is all that matters.  The Royal Flush pays 800.  The Straight Flush pays 50.  ONE Straight Flush is equal to 50 High Pairs or 17 Three of a Kinds or even 2 Quads.  The Royal completely dwarfs them all.  The 1 in 47 chance of hitting the Royal leaves us with an expected value of 18.66 which is well above that of a guaranteed Straight or Flush!

            When designing a game, we can't ignore win frequency.  No one likes to play a game that rarely wins even if the payback is fair.  It just doesn't tend to be fun.  But you can't chase the win frequency the way you must chase the expected value.