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Drawing 2 to RF vs. 1

#1

Greetings :

Yesterday I hit a RF at TP in AC. It was a one card draw. On average I would
say I may get the one card draw maybe once an hour playing maybe 600 to 800
hands. Whereas the two card draw happens much more frequently, naturally.

This being said I have hit the RF more times pulling two cards than
squeezing one. The mathematical question is why is this so? Is it because the
frequency of the chances to get the royal with the two card draw overrides the
better odds of pulling one card instead of two?

The Grumpster

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#2

<<On average I would say I may get the one card draw maybe once an hour
playing maybe 600 to 800 hands. Whereas the two card draw happens much more
frequently, naturally.>>

On average you are dealt four to a royal every 2765 hands and three to a
royal every 60 hands.

<<This being said I have hit the RF more times pulling two cards than
squeezing one. The mathematical question is why is this so? Is it because
the frequency of the chances to get the royal with the two card draw
overrides the better odds of pulling one card instead of two?>

You complete the one-card draw every 47 times and the two-card draw every
1081 times, which if you drew to a three-card royal every time would yield
exactly twice as many royals as the one-card draw. You don't draw to it
every time, but it's more than half the time, so you get more total royals
from the two-card draw.

Cogno

#3

I'll assume my favorite 9/6 JoB game with a max-ER strategy, the
numbers will be slightly different for other games or other
strategies.

I ran through the numbers pretty quickly, so there's a risk that they
might be somewhat inaccurate, though I'd be surprised if they were off
by more than a few percent. Certainly, if you want to strongly rely on
those numbers, have them independently verified.

There are 2598960 possibly deals.

There are 940 ways to be dealt exactly 4 cards to a royal, and 936
ways are played as 4RF (4 are played as SF). Holding 4RF is therefore
a 1-in-2776 event. When holding 4RF, there's a 1-in-47 chance of
hitting RF. That means that being dealt and holding 4RF and drawing RF
is a 1-in-130500 event.

There are 43240 ways to be dealt exactly 3 cards to a royal, and 27480
ways are played as 3RF (that's the number I'm the least confident
about - and the calculation is too long to fit between parentheses).
Holding 3RF is therefore a 1-in-95 event. When holding 3RF, there's a
1-in-1081 chance of hitting RF. That means that being dealt and
holding 3RF and drawing RF is a 1-in-102200 event.

That means that approximately you can expect to hit 25% more royals
from holding 3 cards than from holding 4 cards (for every 4 royals
drawn after holding 4RF, you'll get 5 from holding 3RF).

Approximately, 6% of royals are dealt, 31% come from holding 4RF, 40%
come from holding 3RF,
and the other 23% come from holding 2 cards or fewer.

JBQ

···

On 1/15/06, BANDSTAND54@aol.com <BANDSTAND54@aol.com> wrote:

This being said I have hit the RF more times pulling two cards than
squeezing one. The mathematical question is why is this so? Is it because the
frequency of the chances to get the royal with the two card draw overrides the
better odds of pulling one card instead of two?

#4

The frequencies depend on strategy, and are different for
each game. For jacks or better, RF's are most frequent holding
3 cards (39.52% of all Royals). Holding zero cards is hardest
(0.29%). This is the infamous razgu Royal. Pete R. had two in
one day!!!

   N = 0 N = 1 N = 2 N = 3 N = 4 N = 5
Probability of Royal holding N cards:
9/6 JOB:
    0.29% 3.41% 19.62% 39.52% 30.95% 6.22%
10/7 DB:
    0.14% 3.64% 15.10% 36.90% 36.82% 7.39%

Note that for DB, the probabilities are almost the same for
N=3 and N=4.

As a sanity check, note that 40390 * 4 / 2598960 = 6.22%.
This is the percentage of dealt Royals for JOB.

We have played over 70 Royal cycles lifetime, and only have 60
Royals between us. You can see that we are very short of
dealt Royals and Royals from 4-card holds:

   N = 0 N = 1 N = 2 N = 3 N = 4 N = 5
     0 3 8 35 13 1

Effen

···

--- BANDSTAND54@AOL.COM wrote:

This being said I have hit the RF more times pulling two cards than
squeezing one. The mathematical question is why is this so? Is it
because the
frequency of the chances to get the royal with the two card draw
overrides the
better odds of pulling one card instead of two?

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#5

Actually, 4-card Royals are dealt once every 2825 hands on
non-joker games, so they should happen once every 3-4 hours.

Royal scares happen once every 900-1000 hands depending on
the type of game. However, Lacey and I had 8 natural RF
scares in 1600 hands of $5 JB yesterday. We hit one straight
and one flush, but no SF or RF from these 8 scares.

Effen

···

--- BANDSTAND54@AOL.COM wrote:

Greetings :

Yesterday I hit a RF at TP in AC. It was a one card draw. On average I
would
say I may get the one card draw maybe once an hour playing maybe 600 to
800 hands.

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#6

On average, how many times between royals in 9/6 JOB will the resulting hand be a 4RF? I know to expect 46 non-royals for every royal when dealt 4RF, but what about when including all other dealt hands: 3RF, 2RF, 1RF, and even a hi pair?

  John

···

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#7

The proper way to answer that question would be to perform a computer
analysis - it's too complex (in my opinion) to be practical by hand.

That being said, here are a few numbers, assuming 9/6 JoB with max-EV strategy:

From 4RF, 46 draws out of 47 result in 4RF, i.e. approximately once
every 2836 hands.

From 3RF, 90 draws out of 1081 result in 4RF, i.e. approximately once
every 1081 hands.

From a high pair, there are between 1 and 8 ways to get RF4, depending
on how many RF cards you discarded, out of 16215 possible draws.

There are other hands that could result in 4RF, e.g. a pair of tens,
or some 3SF, or 2 unsuited high cards.

JBQ

···

On 1/15/06, John <john_oc_ca@yahoo.com> wrote:

On average, how many times between royals in 9/6 JOB will the resulting hand be a 4RF? I know to expect 46 non-royals for every royal when dealt 4RF, but what about when including all other dealt hands: 3RF, 2RF, 1RF, and even a hi pair?

  John

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Got holiday prints? See all the ways to get quality prints in your hands ASAP.

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