For the past couple of weeks, I’ve
been discussing how expected value (EV) has shaped the strategies for virtually
every casino game. I’ve covered blackjack and Three Card Poker. But, the
concept of expected value gained popularity as a result of video poker.
I’ve told the story many times
of how 20 years ago, my father, Lenny Frome, saw identical video poker machines
in two different casinos, boasting two completely different paybacks. Knowing
that in Las Vegas any game that uses a deck of cards must play randomly, he knew
something didn’t make sense. So, he sat down to figure it out.
What my father realized was that
without bluffing, tells and live opponents, video poker was a game based 100% on
math and nothing but math. You’re dealt five cards. There are 32 ways to
hold/draw these cards. By calculating the probabilities of the resulting hands
and multiplying by the paytable values, he was able to determine the win-power
of each of these 32 possibilities. The hand with the highest win-power was the
one that maximized your chances of winning in the long term. Eventually, he
decided to call this the expected value instead of win-power.
While this calculation might
sound scary to some, thanks to today’s computers, it really is relatively easy
to compute. While there are a variety of different ways to perform the detailed
work, it still comes down to going through each one of the possible draws and
determining how many winning hand of each type will result.
When the player holds two cards,
there are 16,215 possible draws. When he discards all five cards, there are
1,533,939 possible draws. Using a computer program, we play out each one of
these draws and determine how many of each winning hand there will be. We
multiply the count of each hand type times the payout amount for that hand and
add up these numbers. We then divide by the total number of draws. While this
might sound confusing, you’ll notice that we’re only using basic math —
multiplication, addition and division.
The good news is that the player
really doesn’t have to concern himself with the calculation of the expected
value. My point in presenting it is to show that the concepts of expert strategy
are based in sound mathematical concepts that can be calculated by anybody who
is good with numbers and/or a computer.
Once you understand this, you can
realize that as a player you simply need to play the hand according to which of
the 32 possible plays results in the highest expected value. The better news for
the player is that you don’t really have to go through 32 possible
combinations to figure out which has the highest EV.
For most hands, there is one
obvious way to play it. The rest have just two or three possibilities, and with
a little practice it won’t take long to recognize these situations and learn
to play them correctly.
For example, if you were dealt
the following hand:
4 4 5 6 7
Most people will quickly rule out
30 of the 32 possible ways to play this hand and settle on two realistic
choices. Either hold the pair of 4s or the 4-card straight with no high cards.
Which is the right choice?
In order to answer this, we need
to calculate the expected value for each of these choices. When you hold the low
pair, there are 16,215 possible draws. These draws will lead to 45
four-of-a-kinds, 165 full houses, 1,854 three-of-a-kinds and 2,592 two-pairs.
When we add up the total coins returned (13,356) and divide by the total coins
wagered (16,215), we get an expected value of 0.82.
The 4-card straight is much
simpler to calculation. Of the 47 cards we can draw, eight will result in a
straight. The rest will result in a loss. So, we can expect 32 coins returned
from 47 wagered for an expected value of 0.68.
Once we see the expected values
for each of the two hands, it becomes apparent which is the right play. You
should always hold the low pair when dealt a hand that has a low pair and a
4-card straight. If anyone is wondering, the other 30 possible ways to play this
hand will all result in an expected value of 0.32 or less.
In order to calculate the overall
payback of a video poker machine, we need to look at all 2,598,960 possible
initial deals and all of the possible draws for each one. Fortunately, we have
some shortcuts to do this that help speed up the process.
What happens if you decide to
play a hand in a way that doesn’t maximize the expected value? Well, it really
depends on how big the difference in EV is between the optimal way to play the
hand and the way you choose to play it. Making an occasional error, barring it
being a very ugly one, will really not cost you much. Making the same error over
and over again on a frequently occurring hand can cost you big time.
There are many people who play
blackjack that know to double down on the 10’s and 11’s, but are too timid
to double down on the soft hands. Yes, you’re risking more, but the math tells
us you have an advantage and that’s the right way to play it. There are those
that play too aggressively and perhaps split cards more than they should.
Good blackjack players will stick
to the strategy because they know that is the way to play if you want to win.
Video poker is no different. There are players who play too timidly and those
who play too aggressively. Both sacrifice money because of their styles of play.