# The Penalty Box

In last week's column, I analyzed a particular hand that could be played multiple ways.  The hand was as follows:

J♠        8♦        Q♦       3♥        9♦

From a quick glance, one might think to play the hand as a 4-Card Inside Straight with 2 High Cards, a 3-Card Double Inside Straight Flush with 1 High Card or simply as Two High Cards.  As always, the decision comes down to which of the hands has the highest Expected Value (EV).   In last week's column, instead of simply relying on the EV in a strategy table, I used a program that I created that allows me to put in the EXACT 5 cards and tell it which ones I'm holding and which ones I'm discarding.  It then gives me the exact EV of the hand in question.  Why do I do this instead of just using the value in the strategy table?

The values in the strategy tables are averages of all hands of that particular type.  The accuracy is thus dependent on a few factors, ranging to the nature of the specific hand to the specificity of that hand.  For example, we list the Expected Value of a 4-Card Flush as 1.22.  In reality, there is not a single 4-Card Flush that has that EV.  While there is always the same number of possible ways to draw the Flush (9), the number of High Cards in the hand will impact the exact expected value because it changes the number of ways we can pick up a High Pair.  If we have 0 High Cards, the EV is 1.15.  With 1 High Card it is 1.21 and with 2 High Cards it is 1.28.   We could just as easily list these three hand separately on the strategy table, but it wouldn't change the strategy we would employ at all.  There are no other hands that have an EV between 1.15 and 1.28.  So, in this case we lump all the 4-Card Flushes together and show the average EV for all 85,512 possible 4-Card Flushes.

In a similar fashion, we have a single entry on our Strategy table called the 4-Card Royal which has an expected value of 18.66.  but not all 4-Card Royals are created equal.  We might have 10-J-Q-K which allows for pulling the suited 9 and picking up a Straight Flush.  Or we can pick up an unsuited 9 for a Straight.  However, we also only have 9 ways to pick up a High Pair.  Thus the EV of this hand is rather different from that of J-Q-K-A which has no way to pick up a Straight Flush and also has only one way to pick up a Straight (both ends are NOT open).  But, we get 3 additional cards that will give us the High Pair.

But, there is another item that can affect the specific Expected Value.  What happens if we are dealt a Flush 3-J-Q-K-A.  The Flush has an EV of 6.00 while the 4-Card Royal has an EV of 18.66.  But, when we discard the 3, we lose one opportunity to draw the Flush.  This will certainly NOT drop the EV of the 4-Card Royal to below that of a Flush, but we should recognize the impact of the specific card we discard.  When we discard a card that could help improve the final hand, it is called a 'penalty card'.  In this particular case, there is no impact to our strategy as a result of discarding the 3, so we are safe to lump all 4-Card Royals together.

However, as we go down further on our strategy table, we begin to break apart the hands into smaller groupings.  We don't have all the 4-Card Straights listed together the way we do the 4-Card Flushes.  Because a Straight only pays 4 and there are only 8 ways to complete them, the EV of Straights drops to the point where it is very close to many 3-Card Straight Flushes, 2-Card Royals and even High Card hands.  Many of these hands also tend to overlap a lot, as in the example at the beginning of this article.  The hand is 2 High Cards, a 3-Card Straight Flush and a 4-Card Inside Straight all at the same time.  Slight changes in the hand make up could make it other hands all at the same time.

When a hand overlaps as this one does, there is usually at least some penalty card situations.  In this case, if we choose to play the hand as 2 High Cards, discarding the 8 and 9 create the penalty card situation.  We wouldn't want to draw an 8, 9 and 3, but we wouldn't mind drawing an 8, 9 and 10.  While this may not be the most common outcome, it is one that would complete the Straight and give us one of the highest possible payouts for the 2 High Cards.  So, discarding them may reduce the ACTUAL Expected Value slightly from the one we may find under 2 High Cards in the strategy table.

Likewise, when we hold the 8, 9 and Q, we are discarding the Jack which is a penalty card.  It can be used to complete a Straight or we might pick up another Jack to make a High Pair.  So, I calculate the exact Expected Value in last week's column to make sure the result was 100% accurate.

As I've said many times in my column, you don't need to memorize the Expected Value of any hand because the value itself is meaningless.  What matters is the relative value.  You need to know which hand has the higher EV.   Once in a while, a penalty card situation will cause a hand as it is shown on the strategy table to have an ACTUAL Expected Value that actually drops it to below that of another playable hand from that same 5-card draw.  This in essence creates an exception condition to how the hand should be played when using a strategy table.  The hand should STILL be played according to which has the higher Expected Value, but because we are using the 'average' shown on a strategy table, we don't actually do this.

When my father, Lenny Frome, developed Expert Strategy, he was well aware of this situation.  He felt that the impact on the payback of these exceptions was too small to be concerned with relative to the idea of listing out what could be several to dozens more lines on the strategy table.  Learning Expert Strategy can be enough of a challenge.  He didn't want to complicate it further by trying to list out hands that might look something like this:

·         4-Card Straight with 2 High Cards, EXCEPT if there is a 3-Card Straight Flush, but ONLY if the 2 High Cards are part of the 3-Card Straight Flush

I tend to agree with my father and learning these extra rules are only for diehards and even then, the risk of error might be more than the extra 0.001% it might yield in payback.