Royal deprivation: a coherent answer

Today's QOD was answered in a somewhat rambling way, so:

You should expect a royal roughly every 40,000 hands. Failing to get that royal costs you an 800-unit payout. 800 is 2% of 40,000.

The above is true for almost all variants of video poker. 40,000 royal-less hands of VP should cost you $1000 "extra," over and above the inherent negative EV of the game, at .25 denoms.

You could sit down and get a royal with your first ever-in-your-life hand.  Or you could play and play and play and never get one.

Most people play VP at more than one casino, different machines, so why would "every xx,xxx hands" matter as far as the odds of getting one being touted as a skill set?  My first one was at Circus Circus in Tunica.  Same trip I got two royals at Treasure Bay (Tunica), two different days, two different machines.

I've lost count how many royals I've had, like 30 or more.  I've gotten royals after holding just one card.  No skill there.  I think the least royals for me have come from holding 4 to the royal.  I have NEVER gotten a royal on a re-draw, out of a gazillion re-draws.

Not much skill really, and no units of play will put anybody in line, i.e. "due" for one.  Just hope and luck. 

As always, JMHO.

Candy

Skill determines your base return. That number is always calculated based on perfect play. It has nothing to do with the number of royals you get.

That is point I was trying to make... and obviously failed.  The OP's question, and the answer, concerned how often to expect a royal, how often he "should see" see a royal in X number of hands played.

Candy

That's why "expectation" confuses people. The effect is a convergence of two seemingly opposing phenomena: 1) the more outcomes you have, the closer your result will be to expectation, and 2) the more outcomes you have, the less likely that your result will be exactly at expectation. So if you play 40,000 hands, it's only about a 50% chance that you'll get the expected number of royals (one); but if you play 4,000,000 hands, it's extremely unlikely that you'll get the expected number of royals (100). Yet, you're much more likely to get close to expectation when you play 4,000,000 hands than 40,000.

Head hurt yet?

Ouch!  Yeah.  Good explanation about the statistical difference.

I hit a $2 royal at the Plaza in late March. I hit another $1 royal on a cruise I took in April. On June 3 I hit another $2 royal at the Plaza, maybe at the same machine as my first one. This was a dealt royal, one in 640,000; I'll probably never hit another. The next day I hit a 25-cent royal at The D; the progressive meter paid me an additional $70. I have had streaks with no royals that went on forever. I've never had so many in a short period except perhaps while playing a 50-cent 10-play machine at the Casino Royale, when they had coupons offering double-pay on royals.

One thing about statistics and gambling that most people don't understand is that events meet expectation in the long run but tend to ''clump up" rather than distribute evenly. You'll get a royal about every 40,000 hands over time, but that doesn't mean they'll come discretely, like clockwork, or that they'll be "due" at any given point. So we've all experienced royal bonanzas (O happy day!) and royal droughts (why do I play this stupid game?).