I'm curious to know the vp pro's stance on this subject. Out of all
the literature I've perused the best I could find were statements to
the effect of "the Nevada Gaming Board is very vigilant so folks can
play with peace of mind". So is the standard operating procedure to
simply limit play to well known Nevada casinos? This subject doesn't
even come up in Bob Dancer's "Million Dollar Video Poker" and gets
very scant coverage elsewhere. It stands to reason that well known
Nevada casinos are rational and would not jeopardize their gaming
license with hanky-panky, but what about some of the lesser places?
What about non-Nevada jurisdictions? How do serious players identify
safe places to play? Has any pro ever found himself playing a rigged
machine? How was this determined and what subsequent actions were
taken? These are the sorts of questions I'd like to see addressed. If
anyone knows of websites, news letters and/or books that cover this
subject please post. TIA.
rigged/tampered machines
A lawsuit was brought to the courts in Wisconsin claiming the video
poker machines were purposely set to not deal a Royal Flush. But the
state wouldn't let the suit come to court because all Indian
reservations are sovereign land and can't be taken to court, local or
Federal.
Hard to believe anyone would do such a thing to their customers until
you recall the story of American Coin Company here in Las Vegas that
owned video poker machines in local bars that were found to have a
special R.G. chip that for some strange reason, didn't have a Royal
Flush programmed into it. Even if you had 4 cards to the Royal, it
was impossible to draw the fifth card. They were put out of business
immediately and made to pay the lawsuit. At the Indian casinos, when
the state gaming commission showed up at the front door to check the
machines, they were given free key rings and comps to the buffet, but
nothing more. When they went to Chief No Change to answer for the
casinos refusal to be overseen, he did a rain dance and threw them
both an umbrella and a dream catcher (that was made in China), and
said in his best Klinket "mita wanta getta liffa somewheres elsa."
I knew that American Coin was the company that Frank Romano had been
associated with and that it was involved in the biggest cheating
scandal in Nevada gaming history. In July 1989 the Nevada Gaming
Control Board seized about 1,000 of the company's gaming machines
in
93 southern Nevada locations (mostly bars and taverns) after it
discovered that they contained unapproved computer chips. The
company's video poker machines had been altered to avoid giving a
royal flush and their keno machines had also been programmed to avoid
giving out the top jackpots.
Eventually, the company surrendered its license and paid $1 million
in fines. Authorities also pursued a criminal case against the
company but were later forced to drop those proceedings when their
star witness Larry Volk, the American Coin programmer who said he had
been ordered to program the rigged chips, was shot in the back of the
head and killed outside his Las Vegas mobile home in October 1990. A
suspect was later arrested and tried in the case but a jury
didn't
convict him.
i think you're least likely to get cheated at small casinos in states
with regulated gambling (nevada,nj,miss) because these casinos could
lose their license if they are caught cheating and even though
business isn't what it used to be, most casinos are still very
profitable, so losing a license is a big deal
next in risk would be the big corporate casinos in states with
regulated gambling, the problem is that they are so big that they
will probably never lose their license, case in point the Venetian
rigged drawings - they should have lost their license for that, a
smaller casino would have, but they are so big they did not, they
received a slap on the wrist and the employees involved got jobs in
other states, no one was prosecuted
indian gambling and internet gambling are completely unregulated,
your odds of getting cheated are very high
"lottery style" machines are rigged (non-random) and are legal in
many jurisdictions though they can look just like regular nevada
style video poker machines
igt, ballys, sigma and other nevada licensed machine manufacturers
must ship certified random machines and in nevada they are subject to
spot inspection to insure they have not been tampered with in the
field, however outside of the regulated states there is no guarantee
that these machines are not modified (rigged), modification is fairly
easy, all you need is a software programmer who knows how to modify
rom software code
Interesting that you mention the 4 to the Royal case because that's
exactly what happened to me recently. In fact it was part of my
motivation for this query. Here's the details:
The machine I was playing on was one of the newer TITO 3/5 play IGTs.
I had played about 30000 hands of NSUD. Over the course of these 30000
hands I was dealt zero Royals. I suppose that since I had not
completed a number of hands equivalent to a Royal "cycle" I should not
be surprised by this outcome. However, I was dealt 4 to the Royal 4
times for a total of 20 possible hands (5 play). The only positive
result out of these 20 hands was 1 Wild Royal. That's it. The other 19
were pairs or high card! As this result is significantly below
expectation (a normal result would be 6 or 7 hands resulting in a
straight or better) it had me wondering if an algorithm was at work
for restricting cards in cases such as 4 to the Royal. Although highly
unlikely this result is within the realm of possibility. I am going to
play some more on this machine and will gather more data, but
obviously I can't do that forever. This is why I was curious as to how
the pros approach such a situation.
--- In vpFREE@yahoogroups.com, "dirtyroyal2004a"
<dirtyroyal2004a@y...> wrote:
···
A lawsuit was brought to the courts in Wisconsin claiming the video
poker machines were purposely set to not deal a Royal Flush. But the
state wouldn't let the suit come to court because all Indian
reservations are sovereign land and can't be taken to court, local or
Federal.Hard to believe anyone would do such a thing to their customers until
you recall the story of American Coin Company here in Las Vegas that
owned video poker machines in local bars that were found to have a
special R.G. chip that for some strange reason, didn't have a Royal
Flush programmed into it. Even if you had 4 cards to the Royal, it
was impossible to draw the fifth card. They were put out of business
immediately and made to pay the lawsuit. At the Indian casinos, when
the state gaming commission showed up at the front door to check the
machines, they were given free key rings and comps to the buffet, but
nothing more. When they went to Chief No Change to answer for the
casinos refusal to be overseen, he did a rain dance and threw them
both an umbrella and a dream catcher (that was made in China), and
said in his best Klinket "mita wanta getta liffa somewheres elsa."I knew that American Coin was the company that Frank Romano had been
associated with and that it was involved in the biggest cheating
scandal in Nevada gaming history. In July 1989 the Nevada Gaming
Control Board seized about 1,000 of the company's gaming machines
in
93 southern Nevada locations (mostly bars and taverns) after it
discovered that they contained unapproved computer chips. The
company's video poker machines had been altered to avoid giving a
royal flush and their keno machines had also been programmed to avoid
giving out the top jackpots.Eventually, the company surrendered its license and paid $1 million
in fines. Authorities also pursued a criminal case against the
company but were later forced to drop those proceedings when their
star witness Larry Volk, the American Coin programmer who said he had
been ordered to program the rigged chips, was shot in the back of the
head and killed outside his Las Vegas mobile home in October 1990. A
suspect was later arrested and tried in the case but a jury
didn't
convict him.
Hacking the firmware is not so easy. Reverse engineering the program
from assembly code is a non-trivial resource constrained task. But
given what's at stake someone might just throw the neccessary
resources at it. They might be tempted to do that in a jurisdiction
that has no effective gaming oversight. If gaming manufacturers wanted
to assuage fears about tampering they could make the hardware
virtually tamper proof. But if they are not compelled to do so why
incur the additional manufacturing cost? I'd be very interested in
knowing what state gaming board regulations dictate regarding
deployment of new machines and how field inspections are conducted.
--- In vpFREE@yahoogroups.com, "dirtyroyal2004a"
<dirtyroyal2004a@y...> wrote:
···
i think you're least likely to get cheated at small casinos in states
with regulated gambling (nevada,nj,miss) because these casinos could
lose their license if they are caught cheating and even though
business isn't what it used to be, most casinos are still very
profitable, so losing a license is a big dealnext in risk would be the big corporate casinos in states with
regulated gambling, the problem is that they are so big that they
will probably never lose their license, case in point the Venetian
rigged drawings - they should have lost their license for that, a
smaller casino would have, but they are so big they did not, they
received a slap on the wrist and the employees involved got jobs in
other states, no one was prosecutedindian gambling and internet gambling are completely unregulated,
your odds of getting cheated are very high"lottery style" machines are rigged (non-random) and are legal in
many jurisdictions though they can look just like regular nevada
style video poker machinesigt, ballys, sigma and other nevada licensed machine manufacturers
must ship certified random machines and in nevada they are subject to
spot inspection to insure they have not been tampered with in the
field, however outside of the regulated states there is no guarantee
that these machines are not modified (rigged), modification is fairly
easy, all you need is a software programmer who knows how to modify
rom software code
the probability of playing n hands without hitting a hand that has
probability p is (1-p)^n, for a cycle of a hand you will find this
is .3675 (36.75%), for x number of cycles it is .3675^x
example: in 9/6 jacks the probability of hitting a quad is .00236
(.236%) and the cycle is 423, so:
hands cycles probability of not hitting
423 1 36.75%
846 2 13.51%
1269 3 4.96%
1692 4 1.82%
2116 5 .67%
4233 10 .004%
so if you go 5 cycles, that's very unusual (.67%)
if you go 10 (.004%) most likely something is wrong
--- In vpFREE@yahoogroups.com, "dirtyroyal2004a"
<dirtyroyal2004a@y...> wrote:
"lottery style" machines are rigged (non-random) and are legal in
many jurisdictions though they can look just like regular nevada
style video poker machines
Is this type of "fake VP" legal in Nevada? If so, is there anyway to
differentiate it from a standard VP game?
Also, I think the "rationale" American Coin used to justify rigging
the VP was that most of their customers were small venues, like bars,
and if someone won a big jackpot (or two) late at night, the bar
didn't have enough cash on hand to pay the winner. So, to avoid
losing these small customers, they rigged the machines, hoping it
would not be noticed. That was the wrong way to solve the problem.
I think the same problem exists today, in grocery and drug stores
with a few machines. If someone hits a big jackpot I think (not sure
though) the attendant calls a central office which sends someone out
immediately to pay the jackpot. That appeared to be what I saw the
other day at Smiths when someone got a $7500 RF. A man walked in
with a huge security guy and made the payoff. But I might be
wrong ... maybe they simply restocking the safe with new money.
Does anyone know how big payoffs at small insecure venues (like
Smiths) are handled?
> "lottery style" machines are rigged (non-random) and are legal in
> many jurisdictions though they can look just like regular nevada
> style video poker machines
Is this type of "fake VP" legal in Nevada? If so, is there anyway
to
differentiate it from a standard VP game?
http://www.nigc.gov/nigc/documents/releases/advcomltr-0104-2.jsp
rigged (non-random) machines are called class 2 devices by nigc,
class 3 is nevada style machines (random)
non-random machines are illegal in nevada, in addition no
manufacturer licensed in nevada can produce non-random machines even
for sale in other states, so igt, ballys, sigma, williams do not
legally make non-random machines
···
--- In vpFREE@yahoogroups.com, "brumar_lv" <brumar_lv@y...> wrote:
Thanks for posting this info. Playing All-American, I have gone
18,200 hands and counting without a straight flush. I had been
wondering about the improbability of my dry spell (7.82% probability
of not hitting a STFL over this span, if I applied the formula
correctly). On the flip side of the variance spectrum, on Saturday
I hit my second Royal in the last 200 hands, so I am no longer
complaining about the STFL drought.
By the way, the first Royal last Sunday at Harrah's KC prompted my
post about tipping (I didn't tip at all on that one). The second
one came 15 minutes into my next session--Saturday at the President
(St. Louis)--and based on what was posted here about gambling
etiquette, norms and customs I tipped the two attendants $5 each.
-JAD
--- In vpFREE@yahoogroups.com, "dirtyroyal2004a"
<dirtyroyal2004a@y...> wrote:
···
the probability of playing n hands without hitting a hand that has
probability p is (1-p)^n, for a cycle of a hand you will find this
is .3675 (36.75%), for x number of cycles it is .3675^xexample: in 9/6 jacks the probability of hitting a quad is .00236
(.236%) and the cycle is 423, so:hands cycles probability of not hitting
423 1 36.75%
846 2 13.51%
1269 3 4.96%
1692 4 1.82%
2116 5 .67%
4233 10 .004%so if you go 5 cycles, that's very unusual (.67%)
if you go 10 (.004%) most likely something is wrong
dirtyroyal2004a wrote:
the probability of playing n hands without hitting a hand that has
probability p is (1-p)^n, for a cycle of a hand you will find this
is .3675 (36.75%), for x number of cycles it is .3675^x
It's always interesting to know just what the probability of a current
drought is. By this formula, for example, the probability of a
2-cycle drought is 13.5%.
But since over a long period of play, lengthly droughts can reasonably
be expected, it's always been my keen curiosity what the probability
of a n-cycle drought is over the course of a much larger number of cycles.
···
------
Take, for example, the probability of suffering a painful 5-cycle
drought over the course of 20 cycles of play.
To me, this puts a drought in a much more practical perspective. The
probability of a drought over any 5-cycles is, of course, a slim .7%.
For a royal, that's sufficient to send a player crying to the bank,
bemoaning what a cruel run of luck they've had.
However, if it should turn out that the probability of hitting such a
dry streak over 20 cycles is something like 3%-5%, then that hardly
puts them in rarified company. (Well, if the hand in question is the
royal there is the question of just how many players will stick that
drought out through the full 5 cycles rather than run home with their
tail between their legs.)
------
So my question is whether this math is nearly so readily accessible.
No solution has popped into my relatively simple mind 
Anyone have
some insight to offer?
- Harry
"Harry Porter" <harry.porter@v...> wrote:
Take, for example, the probability of suffering a painful 5-cycle
drought over the course of 20 cycles of play.
unless i'm missing something, it's straightforward:
(4 royals over 20 cycles?)
let Pr be probability of hitting a royal, then royal cycle is 1/Pr
(1-Pr)^(20/Pr -4) x Pr^4 x combinations(20/Pr:4)
combinations(x:y)=x!/y!(x-y)!, !=factorial (3!=3x2x1)
so for Pr=.000025, 1/Pr=40,000
combinations(800,000:4)=
800,000x799,999x799,998x799,997/4x3x2x1=2e22
2e-9 x 4e-19 x 2e22 = 16e-6 = .002%
--- In vpFREE@yahoogroups.com, "dirtyroyal2004a"
<dirtyroyal2004a@y...> wrote:
i think you're least likely to get cheated at small casinos in states
with regulated gambling (nevada,nj,miss) because these casinos could
lose their license if they are caught cheating and even though
business isn't what it used to be, most casinos are still very
profitable, so losing a license is a big dealnext in risk would be the big corporate casinos in states with
regulated gambling, the problem is that they are so big that they
will probably never lose their license, case in point the Venetian
rigged drawings - they should have lost their license for that, a
smaller casino would have, but they are so big they did not, they
received a slap on the wrist and the employees involved got jobs in
other states, no one was prosecuted
With the exception of one casino where adjacent PickEm machines went
BSOD within minutes of each other, I'm comfortable with the Nevada
casinos that I've visited which are listed in the vpFree database.
indian gambling and internet gambling are completely unregulated,
your odds of getting cheated are very high
I completely agree. The literature is rife with scandals at MidWestern
Indian casinos. You pointed out a lawsuit vs. a Wisconsin Indian
casino regarding vp Royals. Not too long ago a Wisconsin Indian casino
(might be the same one, I don't know) was caught removing 10s from
blackjack shoes. I never have, nor will I in the future play at an
Indian casino.
"lottery style" machines are rigged (non-random) and are legal in
many jurisdictions though they can look just like regular nevada
style video poker machinesigt, ballys, sigma and other nevada licensed machine manufacturers
must ship certified random machines and in nevada they are subject to
spot inspection to insure they have not been tampered with in the
field, however outside of the regulated states there is no guarantee
that these machines are not modified (rigged), modification is fairly
easy, all you need is a software programmer who knows how to modify
rom software code
What do you mean by "must ship"? That's easily circumventable and
unenforceable. If it's a state law simply set up manufacturing in a
different state. If it's a federal law (which I doubt) simply
manufacture off shore. The casinos certainly won't complain if the vp
machines roll out of the factory gaffed to begin with as this will
make them more money. The field tampering isn't *that* easy, but is
certainly doable if enough resources are thrown at it (which is a
sentiment echoed in the interview presented at a URL which you gave:
http://www.americancasinoguide.com/Tips/Slots-Honest.shtml). Basically
we have to depend on the gaming law enforcement body of the local
jurisdiction to protect the gambling public. Nevada seems to do a good
job, but I have not yet reached that level of comfort elsewhere. In
fact my query was triggered by an incident with an IGT machine in an
Indiana casino. I have written the Indiana gaming commission, but have
yet to receive an answer. Speaking of IGT, the interview of IGT given
in the above URL certainly doesn't leave the reader with a warm &
fuzzy feeling regarding the integrity of IGT devices.
Regarding the actual probabilities, exploring confidence intervals
certainly lends more credence to a claim that something isn't right,
but it isn't even neccessary to go that far. In my case of being dealt
4 to the Royal 4 times (total of 20 hands) that should produce a
result of straight or better about 6-7 times if probability holds up
(p = .34). 1 positive result out of 20 certainly would strike most
reasonable people as "extremely unlucky" at the very least. In fact
the result falls just outside of 2 cycles. Add to that the facts that
more than the Royal cycle number of hands were played *and* the deuces
cycle was far below expectation (about .5) we have a very suspicious
NSUD machine in my book. I was hoping that someone could assuage my
fears with some convincing statistics, but that hasn't happened yet.
Maybe a thorough review of vp statistics will make me feel better.
"Harry Porter" <harry.porter@v...> wrote:
> Take, for example, the probability of suffering a painful 5-cycle
> drought over the course of 20 cycles of play.
dirtyroyal2004a wrote:
unless i'm missing something, it's straightforward:
(4 royals over 20 cycles?)
let Pr be probability of hitting a royal, then royal cycle is 1/Pr
(1-Pr)^(20/Pr -4) x Pr^4 x combinations(20/Pr:4)
combinations(x:y)=x!/y!(x-y)!, !=factorial (3!=3x2x1)so for Pr=.000025, 1/Pr=40,000
combinations(800,000:4)=
800,000x799,999x799,998x799,997/4x3x2x1=2e22
2e-9 x 4e-19 x 2e22 = 16e-6 = .002%
This misses the mark. What I'm asking is the probability of suffering
an extended 5-cycle drought at some point during 20 cycles of play.
In other words, expressing the problem in terms of royals and assuming
a cycle of 40K hands, how likely is a player to suffer 200K or more
consecutive hands without a royal at some point within a given span of
800K hands.
That's a considerably more expansive problem than looking at the
probability that 4 royals are hit over 20 cycles. (By the way, an
example in which 100 royals are hit during the 800K hands, but none
within a period of 200K hands, would satisfy the criteria.)
Keep in mind the motivation behind the question is to evaluate just
how rare extended droughts actually are, beyond simply calculating
their probability at a single point in time.
If my question isn't clear yet, please ask me to explain in more detail.
- Harry
vp_nbi wrote:
--- In vpFREE@yahoogroups.com, "dirtyroyal2004a"
<dirtyroyal2004a@y...> wrote:
snip
> indian gambling and internet gambling are completely unregulated,
> your odds of getting cheated are very highI completely agree. The literature is rife with scandals at MidWestern
Indian casinos. You pointed out a lawsuit vs. a Wisconsin Indian
casino regarding vp Royals. Not too long ago a Wisconsin Indian casino
(might be the same one, I don't know) was caught removing 10s from
blackjack shoes. I never have, nor will I in the future play at an
Indian casino.
snip
I _CONSTANTLY_ hear that Indian casinos are not trustworthy. I don't know if it is simply because of prejudice, or urban legend, or because of suspicions because they are unregulated (by any state gaming commission or the Feds), or because there is actual evidence that they have a propensity to cheat. If the latter, perhaps that is rationalized as a way to even the score against the 'white man'; I don' know, but that would seem rather dishonorable to me. It is also rather disparaging, and if I were an Indian, then I think I'd be opposed to any 'cheating' for the sake of racial relations, and I'd be fighting to clear that up. But I never hear a word from anyone who would admit to being an Indian and opposed to those practices. What's up with that?
Bill Velek
"Harry Porter" <harry.porter@v...> wrote:
In other words, expressing the problem in terms of royals and
assuming
a cycle of 40K hands, how likely is a player to suffer 200K or more
consecutive hands without a royal at some point within a given span
of
800K hands.
ok, 200k hands without a royal=
(1-.000025)^200k=.0067
the probability of that not happening is:
1-.0067=0.9933
the probability of that not happening four times in a row (200k x 4 =
800k) is:
0.9933^4=.97=97%
so, 3% chance an 800k span of hands contains a royaless string of at
least 200k hands
--- In vpFREE@yahoogroups.com, "dirtyroyal2004a"
<dirtyroyal2004a@y...> wrote:
"Harry Porter" <harry.porter@v...> wrote:
> In other words, expressing the problem in terms of royals and
assuming
> a cycle of 40K hands, how likely is a player to suffer 200K or
more
> consecutive hands without a royal at some point within a given
span
···
of
> 800K hands.ok, 200k hands without a royal=
(1-.000025)^200k=.0067the probability of that not happening is:
1-.0067=0.9933the probability of that not happening four times in a row (200k x 4
=
800k) is:
0.9933^4=.97=97%so, 3% chance an 800k span of hands contains a royaless string of
at
least 200k hands
actually, with four blocks of 200k each, each block without a 200k
royaless span, i've created three seams between the blocks, which
could contain a 200k royaless span, the odds of that happening
is .0067 each seam? that would add another 2%
dirtyroyal2004a wrote:
> ok, 200k hands without a royal=
> (1-.000025)^200k=.0067
>
> the probability of that not happening is:
> 1-.0067=0.9933
>
> the probability of that not happening four times in a row
> (200k x 4 = 800k) is: 0.9933^4=.97=97%
>
> so, 3% chance an 800k span of hands contains a royaless string of
> at least 200k hands
you added in a second post:
···
actually, with four blocks of 200k each, each block without a 200k
royaless span, i've created three seams between the blocks, which
could contain a 200k royaless span, the odds of that happening
is .0067 each seam? that would add another 2%
------
Let me address your two posts separately. I'm with you on the first,
but the real fly in the ointment lies in the second.
The probability (P) that there's no royal over the course of 200K
hands is .0067. P that there's at least one royal = .9933 - Agreed
Your next statement is: P that, over 4 sequences of 200K hands, each
one will contain at least one royal = .9933^4 = .9734 (97%) - Agreed.
But as your subsequent post suggests, you agree that this statement
translates into a P that, over 4 sequences of 200K hands, at least one
will be royalless = 3%.
By itself, this doesn't represent the probability that there will be a
string of 200K hands without a royal within 800K hands of play. It
addresses only 4 of the 600,001 strings of 200K hands to be found in
800K hands. (These strings would be the individual ones commencing
with the 1st hand {1-200,000} through the 600,001st hand
{600,001-800,000}.)
------
repeating your second post:
actually, with four blocks of 200k each, each block without a 200k
royaless span, i've created three seams between the blocks, which
could contain a 200k royaless span, the odds of that happening
is .0067 each seam? that would add another 2%
If I'm on target, there are actually 199,999 strings (spans) that
cross each seam. The ultimate probability needs to account for these.
It would appear that you identify the probability at least one of
these strings crossing a given seam will be royalless is .0067.
That's not immediately apparent to me.
If this is correct, I'd be appreciative if you'd lay out the reasoning
for me (for my sake, don't take any shortcuts). This is the point
where I've stumbled in solving this problem since everything up to
this point has been a no-brainer.
Where I get tripped up is that once there's a possibility that a given
200K string contains a royal, the P that any other coincident string
contains a royal is affected. If a given string is royalless, then
the P that coincident strings are royalless is affected as well. This
gives rise to dependencies that I'm not prepared to account for.
If it's possible to avoid any consideration of "coincident strings"
then one would presumably be home free in arriving at a solution.
By the way, for convenience sake, the final solution should be in a
general form that expresses the P of a royalless string of M hands
over total play of N hands.
- Harry
* Bill Velek wrote: I _CONSTANTLY_ hear that Indian casinos are not
trustworthy. I don't
know if it is simply because of prejudice, or urban legend, or because
of suspicions because they are unregulated (by any state gaming
commission or the Feds), or because there is actual evidence that they
have a propensity to cheat.
As someone who is highly skeptical about non-regulated casinos, I would have
to say my questions are grounded in none of the above. They arise from the
complete lack of independent outside auditing and supervision. Dick Mustain,
as I recall, did point to one states rules vis-à-vis Indian casinos (I
think it was Minnesota). Other states are remarkably not specific on the
subject.
[Non-text portions of this message have been removed]
here's my theory
(not sure it's correct but think it is)
ok, you have two blocks, which contain completely random hands, the
only thing you know, by definition, is that they contain at least one
royal and no royal droughts of 5 cycles or larger
now you put the two blocks together
the question is, what is the probability of a 5 cycle royal drought
from the rightmost royal of block 1 to the leftmost royal of block 2?
isn't it (1-Pr)^(5/Pr)?
or the more general:
probability of a royaless block of length L or greater:
(1-Pr)^L
there is only one royaless block across the seam, and it starts with
the rightmost royal of block one, which i know exists by definition,
therefore P=1, and it ends with the leftmost royal of block two,
ditto, the only question is what is its length
i think (1-Pr)^L is correct
comments?