# Rare Gems - Straight Flushes

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One of the ironies about video poker paytables is that
they don't always reward hands more for being more rare. If I were to ask you which occurs more often
in video poker - a Flush, a Straight or a Full House, I'm guessing most of you
would say a Straight, followed by a Flush and lastly a Full House. It is really a trick question. Without knowing what the paytable is, there
is no way to answer the question accurately.
The only thing we know is that, in general, a Full House outranks a
Flush, which outranks a Straight.

On a full-pay video poker machine, assuming you use
Expert Strategy, you will actually hit more Full Houses than either of the
other two. A Straight will occur just
slightly more often than a Flush. Upon
close inspection, we realize that this is by far a product of the payouts for
each hand than it is a product of the hands themselves. If we take a look at the game of All
American Video Poker - which would appear to now be obsolete - we will see a
very different pattern develop. In All
American, a Straight, Flush and Full House all pay 8. With no reason to go for one or the others,
the pure probabilities of hitting each hand begin to show up. As a result, the frequency of Straights and
Flushes increase dramatically, to the point where they occur nearly twice as
often as a Full House.

A similar phenomenon occurs with a Straight Flush. Generally speaking, it occurs just about 4
times as frequently as a Royal Flush, while paying only 1/16th of the
amount. Or we can look at it the other
way and say that it is more than 20 times as rare as a Four of a Kind while
only paying twice as much. When we throw
in the Bonus Video Pokers, it only looks worse.
This far more rare hand might actually pay LESS than many of the Quads
we can hit, which are far more common.

Of course, I'm wondering how many of you have hit nearly
as many Royal Flushes as you've hit Straight Flushes. I doubt you remember your Straight Flushes as
vividly. Winning $62.50 on a max-coin
quarter machine isn't quite as memorable as a cool $1000, but that isn't my
point. If you use Expert Strategy on a
jacks or better machine, you should hit a Royal every 40,400 hands or so and a
Straight Flush every 9200 hands. The key
phrase is "if you use Expert Strategy." Since most Players, at best, use pieces of
strategy, I'm guessing that the Straight Flush shows up far less often because
the partial Straight Flush is frequently overlooked when the Play.

If dealt the following, what's the right play?

J♠ 8♦ Q♦ 3♥ 9♦

Do you play the 4-Card Inside Straight with 2 High Cards,
the 3-Card Double Inside Straight with 1 High Card or the 2 High Cards? As always, there is just one way to determine
the right play. We go to the Expected
Values of each.

Calculating the Expected Value for the 4-Card Inside
Straight is fairly easy. We can draw the
Straight with 4 cards and we can draw a High Pair with 6 more. This will return 22 units to us. Divide by 47 and we get a result of just
below 0.47. For the other two, I ran
them through a program I have that calculates the exact Expected Value given
the specific discards. The Two High
Cards have an Expected Value of just below 0.50 and the 3-Card Double Inside
Straight Flush has an Expected Value of just below 0.53. This is the proper play.

While the odds of hitting the Straight Flush are 1 in
1081, this is still far greater than hitting it with either of the other two
hands (it is zero in these cases).
Ironically, it is not the tremendous payout of the Straight Flush that
causes us to play the hand this way. By
holding a 3-Card Straight Flush, we give ourselves numerous chances to hit just
Straights and Flushes - a combined 1 in 20 (roughly). Throw in opportunities for Three of a Kind
and Two Pairs and this hand simply beats the others.

Now, no one expects you to calculate the Expected Value
of even the 4-Card Inside Straight on the fly or to carry a small computer to
run my program that calculates the exact Expected Value for each hand. It is much easier to simply use a strategy
table that lists out each playable hand.
If we look up the three hands in
a strategy table, we find a 3-Card Double Inside Straight Flush has an Expected
Value of 0.54, the Two High Cards have an Expected Value of 0.49 and the 4-Card
Inside Straight with 2 High Cards doesn't even make it onto our strategy table
because the Two High Cards always outranks it.
These values are the average of all hands of that type so they don't
always equal the exact Expected Value taken into account the exact discards.

In the end, the frequency of a hand occurring is a
product of the paytable and following the right strategy. If you want to get your share of Straight
Flushes, you can't do a lot about the former, but the latter is fully in your
control.