More About Expected Value

          Maybe somebody beat my father to it, but I think the was the first person to use the expression 'expected value' when referring to gambling options.  The concept pre-dated him as my father was hardly the first person to analyze blackjack.  In blackjack, when a strategy was developed that told the Player to double on an 11 looking into a 10, this decision was based on which of the two options (hit or double) had the higher expected value.  Well, sort of.  Since the wager amounts are not the same when you hit vs. double, this had to be taken into account as well.  In reality, in blackjack, it is not possible for the expected value (without taking into account the wager size) to be higher on a double vs. a hit.  But that's for another time.

            In a game like video poker, using expected value is a simple way of comparing one hand to another.  You make your wager upfront and it doesn't change.  Thus, the wager amount is the same for every hand.  In video poker, this leaves us a 'relatively' easy calculation for the expected value.  We take EVERY possible draw for the way we are thinking about playing the hand and add up the amount paid out for each of those possible draws.  We then divide this amount by the number of possible draws.  This way, if we draw 1 (47 possible ways) or draw 3 (16,215 possible ways), we put the two possibilities on equal footing. 

            When we look at a strategy table for video poker, we find that each possible playable draw hand has a specific expected value associated with it.  To be honest, the actual value is not needed when learning how to play the game.  Knowing the specific expected value of a High Pair is not necessary.  Knowing that it is GREATER than the expected value of a 4-Card Straight or a 4-Card Flush (again without knowing these specific values) is all that is necessary.  We play the hand with the highest expected value.  Thus, only relative position on the table is what needs to be memorized.

            We need to also remember that on the strategy table, the expected value shown is the AVERAGE of all hands meeting the description.  Thus, we show an 18.66 for a 4-Card Royal.  But, not all 4-Card Royals are the same.  A 4-Card Royal consisting of an Ace can only make a Royal and not a Straight Flush.  However, if the 4-Card Royal is 10-J-Q-K, we have the added benefit of being able to make a Straight Flush (with a suited 9) and many more Straights than we can with a J-Q-K-A.  At the same time, we have lost one High Card in this tradeoff.  Worse yet, would be a 10-Q-K-A, which cannot make a Straight Flush and can only make 3 Straights AND consists of only 3 High Cards instead of 4.  When the numbers are all crunched, NO 4-Card Royal may actually have an expected value of exactly 18.66. 

            But, we don't have to break down the 4-Card Royal category any further because no other hand comes close to this expected value.  Thus, even if a 10-Q-K-A 4-Card Royal has an expected value slightly below a J-Q-K-A, since there is no hand that has an expected value between these two hands, we don't need to break down the hand further on our strategy table.

            We find that the same thing happens with a 4-Card Flush.  Some of you may have noticed that when we discuss 4-Card Straights or 3-Card Straight Flushes, we are very concerned about the number of High Cards.  This isn't only because this impacts the specific expected value, but rather it impacts the actual strategy as the expected value changes with the number of High Cards.  With 4-Card Flushes, we don't bother with this.  This is for the same reason as with the 4-Card Royal.  While the expected value of a 4-Card Flush varies greatly depending on the number of High Cards, there are no other hands that have an expected value in the same range.  Thus, even if we have a 4-Card Flush with 0 High Cards, which has an expected value of 1.15 or a 4-Card Flush with 2 High Cards, which has an expected value of 1.28, the strategy is still the same.  There are no hands with an expected value between 1.15 and 1.28.  To bring home the point, a 4-Card Flush with 1 High Card has an expected value of 1.21.  If we look at the strategy table we'll see 1.22 listed as the expected value for a 4-Card Flush.  In reality, NO 4-Card Flush actually has this expected value.  It is just an average of all the 4-Card Flushes with 0, 1 or 2 High Cards.

            Now, in case any of you are wondering about a 4-Card Flush with THREE High Cards, we have a special name for that.  It is called a 3-Card Royal!  This has an expected value of 1.41 and can be found right above the 4-Card Flush.  The only reason this hand is broken out is because a hand might be BOTH a 4-Card Flush and a 3-Card Royal and this is telling us when we have one of these hands, we hold only the 3-Card Royal and discard the 4th suited card.  Ironically, this will lower the expected value of the 3-Card Royal slightly as we will reduce our chances of picking up a Flush.  But, I'll leave this 'penalty' card situation for another day.