The Outer Limits

            Expert Strategy consists of three key components - (1) Know which games to play, (2) know the right strategy for those games and (3) know what to expect.   I find the third one to be the most intriguing.  When it comes to video poker, I realize I'm hardly the only source for video poker information.  There are other writers and there is a variety of practice software out there.  These will all help you to learn the first two.  But, I think it is a combination of my math skill and the ability to explain it to the masses that help to separate me from some of these other sources.  One doesn't really need to know what to expect in order to master the first two, but learning it is what keeps the Player disciplined to stick to the plan.

            There are times when I'm playing video poker that even I start to wonder if the games are rigged.  I know they aren't.  But it just seems like that on some days I get more Razgus than I do High Pairs, when you should be getting four times as many of the latter.  But, just because you're supposed to get four times as many that doesn't mean that while you look at a sample of two hours of play that it is going to be all that close to that amount.

            I don't remember exactly when I was introduced to video poker, but I'd have to guess it was in the early 1990s.  Probably won't be a big surprise to learn it was my dad who first showed it to me.  Up until that point I was mostly a blackjack player.  But, even at $3-$5 a hand, blackjack required a bigger bankroll for the most part and quite frankly, I was getting a little bored with it.  For the next several years, when I came out to Las Vegas, I played video poker.  I was probably coming out to Vegas about 2-3 times a year for a week.  Most nights, I would play video poker for 2-4 hours.  And over several years, my Royal Flush count was a big fat ZERO. 

            Even my father was astounded at how many years I went without hitting one.  This was back when jacks or better was one of the only games (no Bonus Versions) and without the Royal, it was hard to have a big winning night.  Adding insult to injury, my now ex-wife, came out to Vegas with me in those early years, and hit a Royal on her very first time playing.  She won $12.50 (she played one nickel).  It took until 1997 until I finally hit one.  My best guess was that I had played well over 100,000 hands without hitting one. 

            As is always the case, it was feast or famine.  I spent two different weeks in Las Vegas over a 6-week period.  In that time, I hit THREE Royal Flushes.  These were all max-coin quarters, so I won $1000 for each.  Again, it is hard to say exactly how many hands I played over this period, but I would guesstimate 15,000 hands and now I had Three Royals.

            So, is the math all wrong.  How could I go over 100,000 hands without hitting one and then hit 3 in 15,000 hands.  I've been asked questions like this so many times over the years.  This past week, I had to answer one about a sidebet for Ultimate Texas Hold'em and a casino that was concerned that the Royal was hitting too often.  It had hit 5 times in a several week period and based on the number of hands played in that period, it clearly 'should not have' hit that often. 

            This concept could be one of the most baffling for even the industry to fully understand.  If some event should occur 1 in 100,000 it does NOT mean that if 2 of them occur over that period that this was some sort of 10 billion to 1 shot.  And even if you get 2 of them over only 25000 hands, it doesn't mean that something is broken.  It means that things are behaving exactly as they are supposed to.

            Let's take a closer look at the examples I cited in this column.  What is the probability that a Player would go 100,000 hands (assuming Expert Play) without hitting a Royal Flush.  The probability of hitting a Royal is 1 in 40,120.  Calculating the probability of NOT hitting one in 100,000 hands requires us to take the 'opposite' probability (40,119 out of 40,120) and multiply it by itself 100,000 times.  This comes out to 8.27%.  Um.... this doesn't sound so rare.  The probability of being dealt a Two Pair or better in a 5-card deal is 7.73%.  No one freaks out when dealt Trips, so no one should question whether or not a machine is broken if no Royal occurs over 100,000 hands (and this was obviously dozens of different machines).  Even if I take it to 150,000 hands, it is still a 2.38% probability.  Luck was not on my side at this point, but the game was just being itself.

            Next up, is hitting three Royals over 15,000 hands.  This calculation is a bit more complex, so I'm not going to spell it all out here at this time.  You'll have to take my word for it that it works out to be 0.60%.  This is 1 in 167.  Given how many video poker Players there are in the world, this 3 out of 15000 has occurred countless times and all within the natural order of things.  Nothing is broken.  Nothing is wrong about the math.  For those interested, the probability of hitting 0 over this period would be 68.81%.  Hitting 1 would be 25.73%.  Hitting 2 would be 4.81%.   That 0.60% is about the same probability as being dealt a Straight or a Flush right off the deal in a 5-card stud game.  It doesn't happen often, but it does happen.  If you've played video poker for a few hours, you know it happens.  It doesn't seem out of the ordinary when it does.  Neither is hitting 3 Royals over 15,000 hands.  You just remember it a lot more because it pays $1000 and not 4 or 6 units per unit wagered.

            The last example was 5 Royal Flushes occurring in about 28,200 hands in a 7-card deal.  A Royal should occur about 1 in 30,940 so having 5 of them occur over a slightly shorter cycle must mean someone is cheating!?  Or maybe not.  The probability of 5 over this period is 0.2181 or about 1 in 458.  I'm not sure exactly how many UTH tables there are, but if there are 1000 of them and we looked at the last 6 weeks of play (or 28,200 hands to be more precise) we would find that (on average) TWO of them would have had five Royals over that time.  It's not a lot.  Far more of them will have the more expected 0 or 1 (these should account for 75% of the tables).

            The example I always love to cite is the frequency of a number repeating THREE times in roulette.  The odds of this (single 0) is 1 in 1369.  I've never seen it happen.  But given that there are dozens of of roulette wheels in Las Vegas, it likely happens here daily 1 or 2 times (maybe more as I'd have to figure out exactly how many spins occur each day here).  If a wheel has 500 spins per day and it happens twice one day, it doesn't mean that the wheel needs maintenance.  It means that even the outliers are going to happen.