# The Penalty Box

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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.