First of all, I have to warn you -- I'm both a math guy and a
compulsive scorekeeper so if you're bored by numbers, you may want to
stop reading right now. In fact, my wife claims that the only reason
I play VP is so that I'll have an excuse to create a spreadsheet.
That, however, is only partially true.
2008, what a frustrating year for VP! My wife and I play Full Pay
Deuces Wild at Red Rock every day. The year started off great. In
the first 3 months of the year, we played 486k hands and had a 2.22%
profit-- nearly triple the expected profit margin of .76%. This was
going to be a very good year ... we thought. Then the bottom dropped
out. In the last 9 months of 2008, we played 934k hands with a loss
of .52%. That may not sound like much but it's more than 1-1/4%
below expected with well over $1 million run through the machine (you
do the math).
What, I wondered, is the probability of showing that much of a loss
for that many hands? I was able to use the bankroll calculation
feature of Video Poker for Winners to determine that the probability
is .62%. In fact, there's only about a 7% probability of showing any
loss at all in that many hands.
So, what happened? Why such poor results for a 9-month period? The
answer, of course: royal flushes (or lack thereof). As I said, we
got off to a great start for the first 3 months of the year. In
January, we had 7 royals in 2.9 RCs (royal cycles = 45,282 hands).
In February we had 7 more royals in 3.8 RCs. And, in March we had 5
royals in 4.1 RCs. We knew we were seriously overachieving with 19
royals in 10.8 RCs but it was fun while it lasted.
What are the odds, you may ask, of getting at least 5 royals in 3
consecutive months playing the number of hands we played each month?
The answer turns out to be 2.07% so that was a very lucky stretch.
But then, we underachieved for the first time in April when we had
only 2 royals in 3.5 RCs. We were out of town for most of the month
of May but managed to hit 1 royal in only .5 RCs. The real shocker
came in June when we had 0 royals in 2.4 RCs. July wasn't very good
either with 2 royals in 3.4 RCs. Then another really bad month in
August with 1 royal in 3.0 RCs. September wasn't too bad with 2
royals in 2.2 RCs. Another bad month in October with 1 royal in 2.1
RCs. November was OK with 2 royals in 1.8 RCs. We had hopes of
ending the year well but had another 0 royal month in December
playing 1.8 RCs.
So, from April through December, just 11 royals in 20.7 RCs. That
was bad enough but the thing that really struck me was the
consistency of the poor results -- we never had more than 2 royals in
any of the 9 months. The probability of getting 2 or fewer royals in
9 consecutive months playing the number of hands we played each month
is .61%. Note that this is virtually the same number as the
probability of losing at the percentage rate we experienced which is
not surprising since we're looking at 2 different ways to express
essentially the same (poor) result. In contrast, if we expand to 3
or fewer royals for the 9 months, the probability jumps to 8.56%.
Finally, one more disturbing statistic. Looking back at the royals
that we did get, only one of them resulted from holding 4 and drawing
one. No wonder we were underachieving on royals -- one royal when
drawing to 4-to-a-royal in 934k hands! I calculate that you should
be dealt 4-to-a-royal (without the deuce) once every 3,022 hands on
average. So that means we had one success in a little over 300 draws.
By the way, we are currently at 125k hands (almost 3 RCs) since our
last royal so our results are actually getting worse (if that's
possible).
Back around September or so, having hit just 3 royals in the
preceding 3 months, my wife asked me if I was sure they hadn't "done
something" to the machines. I laughed and told her that they
couldn't do that because it would be illegal. I'm here to tell you
that I'm not laughing any more. I don't know about you but when
we're getting results that are way less than 1% probable and
approaching 3 standard deviations below the mean, I'm having trouble
attributing it to "a little run of bad luck."
(If we go back far enough - 23 years - we are about 1 mil ahead and so probably should never complain about losing spells that are "only" 8 months long!)