Royal Flush and others Frequency

[ecxpa]

Does the frequency of hitting a Royal Flush or 4 aces with a kicker change with paytables. In other words does the Royal occur just as frequent on 8/5 as it does on 9/6

[onemoretry]

The frequency could change if the differing paytables suggest changing the playing strategy for certain hands, i.e., going for the big hand either more or less often.

[billryan]

   To the best of my knowledge, a person playing perfect strategy will get a RF just as frequently playing a 8/5 version of a game or the 9/6.
Someone who disregards strategy and goes for a RF whenever possible, even if it means just holding a Ten will increase his frequency of getting one but almost certainly lose much, much more.

[New2vp]

In talking Royals, I think there is a decided advantage in the frequency when playing 8/5 rather than 9/6 for the most popular games.  Of course, that is at the expense of other scoring hands, since 9/6 pays 2.2% better than 8/5 rather consistently when all other pay lines are equal.I think the primary reason for more royals is the relatively common deal of 3 Royal cards with another card of the same suit.  There is a tendency for perfect strategy to favor holding 4 cards to the flush when it pays 6.  With a flush counting only 5, there is a larger tendency to hold only 3 to the Royal.The numbers:  Jacks or Better, 0.5% more royals for 8/5 than 9/6  DDB, 1.8% more royals for 8/5 than 9/6  Bonus Deluxe, 3.8% more royals for 8/5 than 9/6  Double Bonus, 1.4% more royals for 8/5/4 than 9/6/4*  Triple Double Bonus, 4.8% more royals for 8/5 than 9/6  Bonus Poker, 0.4% more royals for 8/5 than 9/6***Not the most popular pay schedules for DB, but I have never seen 8/5/5 to compare to 9/6/5 and I was keeping all the pay lines equal.**No such thing as 9/6 Bonus Poker, at least not with the classic paying 2 for 1 for 2 Pair, but I could still make a theoretical comparison.As for Aces with kicker, the results are less pronounced and mixed.  Only two of the games in the comparison above have a special payment for AWAKs.TDB gets a scant 0.25% more AWAKs with 8/5 strategy over 9/6, but DDB gets a microscopic 0.02% fewer AWAKs with 8/5 over 9/6.As onemoretry said, it is the strategy change that accounts for more or fewer final hands.  And billryan's gut instincts are reasonably close within rounding as the percent changes when applied to the tiny probability of getting a royal, while slightly favoring 8/5 pay schedules, are quite minuscule in terms of extra royals per hour (or, more realistically, per week or month).