Penny ante poker!

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One of the first times I ever visited Las Vegas, I was told
about the penny slot machines at the Union Plaza. Of course, we took the trip
downtown and just had to play them. I doubt they paid better than 90 percent or
so, but so long as you dropped a couple of bucks into them, you could go home
and tell everyone how you played penny slot machines.

Over the years, Las Vegas’ reputation for being a low
minimum place to gamble has in many ways disappeared. I suppose it’s still
better than places elsewhere in the country. Surely, finding a seat at a $5
blackjack table isn’t that hard in most casinos. Here, on the East Coast, it
can be difficult to find a $10 seat. Nickel machines have always been plentiful,
but as I took each trip, I almost expected those to go the way of the $2
blackjack table.

What a shock it was on my last trip to find penny
slots in almost every casino I went to. Not just slots, but penny video poker!
Unbelievable! I suppose the advances in technology made this phenomenon
possible.

Now that most machines in Las Vegas are multi-denominational,
casinos didn’t have to dedicate a machine to just pennies. Combine that with
the ability of the machines to take in bills, keep track of credits and most
importantly, print a slip of paper with your payout instead of clanking away
pennies, and I guess it’s not so surprising to find penny machines.

Of course, when you combine a penny machine with a 100-play
version of video poker, you no longer really have penny video poker. Playing
just a penny per hand makes you a $1 player, which puts you just below a quarter
max-coin player. Should you decide to double up to 2 pennies, you’re now
wagering more than that quarter max-coin player. That’s not to say that the
experiences are even remotely the same. My wife and I played for two-three hours
on our last night in Las Vegas on a penny 100-play machine having each put in
$20. When the night was done, I think we were down about $3. Not bad for all
that entertainment!

Sitting there for those hours playing, a lot of questions
popped into my head as I watched (and listened) to the machine play through the
100 draws. Every time I had a 4-card flush or a 4-card straight, I could quickly
figure out the number of ”˜hits’ (i.e. completing the flush or straight) I
expected on average. What I didn’t know was, when I only drew 10 or 12 flushes
(instead of the average of 19-20), was the machine behaving “oddly” or
was I just dealing with the normal flows of random numbers.

Similar questions popped into my head for a lot of other
hands, ranging from low pairs to high pairs to three of a kinds. As I’ve
stated many times, knowing what to expect is an integral part of Expert
Strategy. Knowing what the average number of hits is, is definitely
important, but it seemed to me that knowing how likely other results were, was
equally important.

So, when I got home, I started to work on the problem.
Fortunately, with the power of computers, it’s a pretty easy problem to solve.
It should be noted, that it’s possible to calculate all this purely
mathematically. However, I believe that if I start explaining bell curves and
standard deviations, most of you will either fall asleep, turn the page, or have
dangerous flashbacks to math class of long ago! I prefer the method that keeps
things in simple terms for everyone to understand. The results may not be quite
as accurate, but they will be close enough for these purposes. Besides, results
are only useful if they can be understood by the people who need to use them!

Basically, I created a simple program in which I can set up
any pre-draw hand. In this case, I gave it a three of a kind. I then had it draw
two cards 100 times for each of the 100 hands. I counted up the number of full
houses and four of a kinds that were drawn for each of the 100 draws. I ran this
process 100,000 times and kept a count of how many times the program wound up
with each unique total. The table of the results is presented below:

Mathematically, on average, we would expect to get about 4.25
four of a kinds per 100 draws. The table above pretty much shows that, with 4
being the most common single result, occurring about 20 percent of the time.

As should be no surprise, as the results move away from 4 in
both directions, higher and lower, the results become less common, but even at
zero four of a kinds, we are still talking about 1.3 percent of the time. Nobody
wants that result, but we’re not talking astronomical odds here. The good news
is that it’s not all that rare to hit 6, 7, 8, 9 or once in a while even more
quads when starting with three of a kind.

Full Houses on average should occur about 6.1 times per 100
draws of this type. Again, our table shows the highest concentration right where
it belongs, on the 6. In this case, the distribution is a little wider. Only
16.64 percent will be right on the 6, on average. Winding up with 5 or 7 full
houses is almost as likely as 6. Winding up with no full houses is much less
likely than zero four of a kinds, but we’re still in the realm of reasonably
possible.

For the purposes of this calculation, to keep the processing
simple, I calculated full houses and four of a kinds completely independent of
each other within the 100 draws. In reality, there will be a small correlation
between the two.


Within the 100 draws, if there is an exceptionally large
number of full houses, there will be a slightly lesser number of four of a kinds
because there are fewer remaining hands on average. For the purposes of
understanding what to expect, the impact is insignificant.

Tables such as these can be created for every pre-draw hand.
I think that playing 100-play video poker can give a lot of insight into the ups
and downs of a random machine.

I’m frequently asked about why a video poker machine has
hot and cold streaks if it’s random. From the above tables, you can see how it’s
equally (roughly) likely to draw 9 four of a kinds from three of a kind as it is
to draw zero four of a Kinds. The former would be a hot streak and the latter
would be a cold streak. In the grand scheme, it would just be normal behavior
for a random event.

Elliot Frome is a 2nd generation gaming author and
analyst. His father, Lenny Frome was considered one of the premier authors of
Video Poker books. Titles include, Expert Video Poker for Las Vegas
(recently updated for 2003!) and Winning Strategies for Video Poker,
which includes the strategy tables for 61 of the country’s most popular
versions of Video Poker, and the just released Expert Strategy for Three Card
Poker
. Check out Compu-Flyers website at http://www.vpheaven.com for their
full product catalog, or drop Elliot an e-mail at [email protected]

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