What's the Most Common Sequence to a Royal Flush?

[Barry_Glazer]

Thanks for the math -- getting dealt three is my most common way there, with two next, and apparently with good basis, although I should be hitting a 4-to-a-royal more often, apparently.

I was curious (is my strategy card wrong?) about the remark of holding a King and tossing the suited 10 -- I assume you mean when there is another non-suited high (face) card to hold with the King, as opposed to the Ace, where you only hold the Ace? On my card, suited K-10 is better than K alone.

--BG

···

============

7b. Re: What's the Most Common Sequence to a Royal Flush?
Date: Sat Nov 7, 2009 11:58 pm ((PST))

On Sat, Nov 7, 2009 at 4:09 PM, peppermillionaire > wrote:
> Seems like when I get a royal, it's most often when I get dealt three to the
royal and redraw the other two. Has anyone worked out what sequence is most
likely, and how to do the math on that question? My gut tells me the order is
like this, from most common to least:
>
> Dealt 3, Redealt 2
> Dealt 4, Redealt 1
> Dealt 2, Redealt 3
> Dealt 1, Redealt 4
> Dealt 5
> Dealt 0, Redealt 5

my gut says that's correct for jacks-based games, but that it depends
on the game. i don't think you'll be getting many draw-4 royals
playing deuces wild. at least, i hope not.

for 9/6 jacks:

dealt 5: 4 appearances, hits always, 4 total
dealt 4: 936 appearances, hits 1/47, 19.91 total
dealt 3: 28536 appearances, hits 1/combin(47,2) = 1/1081, 26.40 total
dealt 2: 166380 appearances, hits 1/combin(47,3) = 1/16215, 10.26 total
dealt 1: 402528 appearances, hits 1/combin(47,4) = 1/178365, 2.26 total
dealt 0: 84360 appearances, hits 4/combin(47,5) = 4/1533939 (since
there are four possible royal redraws), 0.055 total

"appearances" info is from frugal, the rest is just math. it's not
perfect, since sometimes you'll draw five and throw away a ten so
there are only three possible royal redraws, or sometimes you'll hold
an ace or king and throw away the suited ten making a royal
impossible, but close enough.

[fivespot]

no, i meant holding the king only while throwing away the suited ten.
that part of my comment was in error, since we're talking about JoB,
where none of the strategy appearances for king-only involve throwing
away a suited ten, as you correctly point out. every time the strategy
tells you to hold a lone king, a royal is possible.

(if we adjust for optimal play using penalty cards, we'd actually
slightly discount the number of royals from drawing three, since
sometimes you don't hold KTs or QTs or JTs when the strategy tells you
to... and from drawing two, since sometimes you don't hold AXTs when
the strategy tells you to... but as with most things involving penalty
cards, the adjustments aren't enough to make any difference worth
caring about. )

cheers,

five

···

On Sun, Nov 8, 2009 at 5:00 AM, Barry Glazer <b.glazer@att.net> wrote:

I was curious (is my strategy card wrong?) about the remark of holding a King and tossing the suited 10 -- I assume you mean when there is another non-suited high (face) card to hold with the King, as opposed to the Ace, where you only hold the Ace? On my card, suited K-10 is better than K alone.