Four to a Royal Promos

[bluestreak86001]

In a message dated 10/4/07 8:34:23 PM US Mountain Standard Time,
mickeycrimm@yahoo.com writes:

···

--- In vpFREE@yahoogroups.com, "drich295" <drichards@...> wrote:

>
>
> Four to a royal occurs about every 460 hands when playing 9/6 JOB
> properly. A little less often than a four of a kind.
>

If I remember correctly, in the Tuscany 4 to the royal promotion, it
had to be a non-paying hand. Example: If your final hand was Ah-Kh-Jh-
Th-4c you got the bonus, but if the final hand was Ah-Kh-Jh-Th-4h you
did not get the bonus because it is a paying hand.

Does your statistic (460 hands) reflect this? Is that how often you
would make a non-paying 4 card royal or does it include all 4 card
royals?

*****
That rule of a non-paying hand certainly cuts the potential value of a four
to a royal promo. Looks like 20-23 is the range of the 47 cards left that will
disqualify the hand. So is that 460 hands any hand that has four to a royal
or just non-paying ones?

BS<BR><BR><BR>**************************************<BR> See what's new at http://www.aol.com</HTML>

[Non-text portions of this message have been removed]

[Mike_Van_Emmerik]

That rule of a non-paying hand certainly cuts the potential value of

a four

to a royal promo. Looks like 20-23 is the range of the 47 cards

left that will

disqualify the hand.

It looks to be 22-23 to me:

(a) KQJTc:
3x9 or 3xA will make a straight (might have discarded one)
2c-8c will make flush (might have discarded one)
high pair: 3xK, 3xQ, 3xJ (might have discarded one)
the 9 of clubs will complete a straight flush (would not have broken that)
Ac will make the royal, but then we don't need this stuff. Let's just
consider the 46 remaining cards that don't make the royal.
Total: 6+7+9+1=23, 22 if broke a straight/flush/high pair
There are similarly 23 of the 46 (exactly half) where the hand is made
as dealt. So half the time it's 23 that will end up paying, the other
half 22, for an average of 22.5.

(b) AKQJc:
3xT will make straight
2c-9c will make flush
3xA, 3xK, 3xQ, 3xJ will make high pair
Total: 3+8+12 = 23, 22 if broke straight/flush/high pair
Similarly, an average of 22.5.

(c) Other ace high, e.g. AKQTc:
3xJ (etc) will make straight
2c-9c will make flush
3xA, 3xK, 3xQ will make high pair
Total: 3+8+12 = 23, 22 if broke a straight/flush/high pair
Similarly, an average of 22.5.

Although there are 3 suits for the discarded card (e.g. Jh, Jd, Js),
you only discard one of them (say Jh), and you can still draw one of
the other two (say Jd) to make a high pair. Also, the "might have
discarded" cards don't overlap, so even though there are three classes
of card you might have thrown away, whatever you discard is a member
of only one of those classes (e.g. it make a straight, or it made a
flush, or it made a high pair; never more than one of these). The 9 of
club is a special case where it makes a straight and a flush, but then
you have a hand worth 50, presumably way more than a 4-card royal is
worth (some 20), so you would not discard the 9c, and all this doesn't
apply.

Obviously, the same goes for the other suits; I just used clubs as a
concrete example.

So it's 23 half of the time, and 22 for the cases where you break a
made hand (let's hope that *they* are not disqualified as well... I
don't think so, this is supposed to be compensation for a "bad beat",
and throwing away a flush to get nothing sounds like a bad beat to
me). That's very close to half the possible draws (actually exactly
half since the Ac completes a royal, and a royal is most certainly a
paying hand, so really it's 23.5 out of 47, not 22.5 out of 46). Four
to the straight flush is such a damned strong hand

M.

[Mike_Van_Emmerik]

Duh. I've done it again. I can't seem to get it through my thick head
that these are *post draw* 4 to the royals.

So there is no adjustment for straight flushes; a straight flush is a
paying hand. Period.

There may be strategy tweaks that affect the frequency of completing
to 4 card royals (the 460 figure). The question is, how often are
post-draw 4 card royals paying hands, thus disqualifying them from the
promotion (if that's how it is worded).

There are 46 possible rank and suit combinations for the final card.
(The 47th makes a 5-card royal.) Correct me if I'm wrong, but I
believe that they are all equally likely. In my previous post, I
showed that there are 23 of the 46 possible draws that end up as
paying hands. The vast majority of these pay way less than 20 bets,
making only high pairs, straights, or flushes, which makes them
irritating but that's irrelevant to the analysis.

So exactly half of post-draw 4-card royal flushes are paying hands. So
the promotion requirement that the hand is non-paying cuts the
expected return of the promotion exactly in half.

Sorry for the confusion.

M.

···

--- In vpFREE@yahoogroups.com, "Mike Van Emmerik" <emmerik@...> wrote:

> That rule of a non-paying hand certainly cuts the potential value
> of a four to a royal promo. Looks like 20-23 is the range of
> the 47 cards left that will disqualify the hand.

It looks to be 22-23 to me:
...