Stddev for # of hands between royals

I play bonus poker. A royal shows up, on average, about little over 40,000 hands. What's the standard deviation for the number of hands for getting a royal?

For example, if it's been 60,000 hands since my last royal, I'd say that's variance, and it'd probably fall within one standard deviation. However, if it's been 500,000 hands since my last royal, that's likely be at the tail end of the distribution.

Just wondering. Some people on another forum had no concept of statistics, and said things like, "Theory is often different from the reality." Umm.... no. That's not how statistics works.

This appears to be a Geometric Distribution problem with p = 1/40000 (or whatever the probability of the Royal is for this machine).

http://en.wikipedia.org/wiki/Geometric_distribution

For this distribution --

mean = 1/p

var = (1-p)/p^2.

std dev = sqrt(var)

In this case, with p = 0.000025, you get

std dev = 39999.5

Note that as p gets close to 0, the standard deviation and mean converge to the same value.

Ok. I really don't know what std deviation means in simple layman's terms. I'm quick with the simple math, but sadly a HS dropout.

I play fpdw optimally. It's been over 235k hands since I hit a royal. Avg occurrence is around 45k hands.

What's my std deviation? Should I really care?

According to my math, [which may very well not be correct], I got a StdDev of 49999.499 [or 50K]. Putting you at 3.7 standard deviations. Not sure if that's how it works, but...that's what I figured.


In any case, that's out of the ordinary. But, you also gotta take into account -- how much do you play? How many hands have you played lifetime? If you've played 10 million hands lifetime, then having such a streak is not so far-fetched. If you've only played 250K hands, then it's a bit more far fetched.


Let's say you flip a coin a bunch of times and you get 20 heads in a row (ie: no-royal-streak) at some point. If you only flipped the coin 100 times, then that 20-in-a-row is out of the ordinary. If you flipped the coin 10,000,000 times, then that 20-streak is not out of the ordinary.



I'm just going to take a wild stab in the dark and guess you haven't played millions of hands of VP.

Although your results are still very possible, you might want to take a look at some other things. First of all, I'd make sure the 235K hands is correct. Usually when counting # of hands played, I'd do something like figure out how many points I earned then divide by $-coin in / point. (ie: At Harrah's it's $10-coin in = 1 point). Some places, like Stations Casinos, have some weird $-coin in / point, especially on those FPDW machines. Hopefully that's the case and you're just at 100K hands without a royal.

If you got that 235K hands and that is accurate, then next I would make sure to be playing the proper strategy. I don't know how playing improper strategy affects the chance of getting a royal. But make sure you're playing properly. ie: With a dealt straight but 4 of the cards are royal cards, make sure you're dumping the straight and going for the royal. Same with Dealt Flush vs 4-to-royal....or 3-to-royal vs pair...etc. I recommend VPW (Video Poker for Winners) to make sure I'm playing the proper strategy.

Lastly, and although this is kind of a stretch, there is a remote possibility the machine is gaffed. Don't have any advice on this one, other than this is likely not the cause. =\

I take it part of your response was directed to my query.

Yes it's stations and it's $12 per point. So very easy to tell. I've certainly played over 2m hand's lifetime and wouldn't be surprised if it was 5m. But I didn't keep records before.

Yes, I play fpdw optimally. Read the books. Used to practice on computer and now practice on cell with a perfect strategy app.

I once spoke to a regular that plays perfect and he stated he once went 9 cycles without a royal. Then hit 7 in a month and 12 in 60 days. Also fpdw. So I know it can get worse. And it can get better!

Quote: RS

According to my math, [which may very well not be correct], I got a StdDev of 49999.499 [or 50K]. Putting you at 3.7 standard deviations. Not sure if that's how it works, but...that's what I figured.

really does not matter what you come up with
the variance is meaningless for what the OP wants

he wants to use the normal distribution
but asks a question that is about the geometric distribution - the wait time

meaningless values

look it up

Quote: RS

Let's say you flip a coin a bunch of times and you get 20 heads in a row (ie: no-royal-streak) at some point.
If you only flipped the coin 100 times, then that 20-in-a-row is out of the ordinary.
If you flipped the coin 10,000,000 times, then that 20-streak is not out of the ordinary.

I agree that the number of trials is important
given enough time and trials,
most all events with a chance greater than 0 can and will happen and should be expected to happen.

ands
the chance of NO 20 run in 10 million flips is abouts 1 in 118 (0.008493878)
so it would not be surprising to never see such a streak over that many trials

but could get close with a 17 run (1 in 68,695,367,949.03 of no run)
or
18 in a row (1 in 192,215,617.98 of no run, ev abouts 19)
or evens
19 in a row (1 in 13,861.97 of no run, ev abouts 10)

the expected # of runs = 4.768 for a run of 20
not a very large number

to the OP
I would just use the distribution for a Royal success to get any confidence interval
40K +/- 40k means nothing (1 SD)

never mix distributions up (a major fire hazard)
and it also makes it look like you do not know what you are talking about
that is OK for most (like me)

Mully

Quote: debitncredit

For example, if it's been 60,000 hands since my last royal <snip>

say the one hand chance of NOT getting a RF = 39999/40000
multiply that by itself 60,000 times and you get
0.22312597642742072415123237831663 (a little less than 1 in 5)
so on average
1 in 5 RF require more than 60,000 hands to show one time

how about N=80k?
N=100k?
(39,999/40000)^N

80k: 0.1353318998404343512411703599501 or abouts 1 in 7
will take longer than 80k hands played (it is the distribution)

100k: 0.08208243346501987752074610972255
1 in 12 abouts

for 500k
now
1 in 268,379.2 would require that many hands played (on average of course)
=============
added
a table of data
the median = 50%
27,726 hands
meaning 50% of RF occur in 27,726 hands of less (at 1 in 40,000 of course)

N	Cumulative
4,214.37	10.0%
6,500.68	15.0%
8,925.63	20.0%
11,507.14	25.0%
14,266.82	30.0%
17,231.10	35.0%
20,432.77	40.0%
23,913.18	45.0%
27,725.54	50.0%
31,939.91	55.0%
36,651.17	60.0%
41,992.36	65.0%
48,158.31	70.0%
55,451.08	75.0%
64,376.71	80.0%
75,883.85	85.0%
92,102.25	90.0%
119,827.79	95.0%
128,753.42	96.0%
140,260.56	97.0%
156,478.96	98.0%
184,204.50	99.0%
188,418.87	99.1%
193,130.14	99.2%
198,471.32	99.3%
204,637.27	99.4%
211,930.05	99.5%
220,855.68	99.6%
232,362.82	99.7%
248,581.22	99.8%
276,306.76	99.9%

========================
your mileage will vary of course between RF
but the distribution shall remain the same
and nothing one can do to change that
unless you go for RF more often than the strat dictates, that is
you want an average of 1 in 25,000


of course, video poker machines have been rigged to pay out less number of RFs in the past (way back in the late 1980s - American Coin is what I read)
and that could be happening now days, any place and any time
and maybe no one even knows if still happens

well, finally
again, do not use standard deviation and the normal distribution for this type of question

Sally

Quote: teliot

This appears to be a Geometric Distribution

What you are looking for is called a "Poison distribution", i.e. the distribution of rare events.

For a Poisson distribution, stddev = mean, hence the stddev is 40000 hands. No need for any calculation.

For royals in single line video poker you can use the exponential distribution to estimate the probability since the geometric distribution converges to exponential pretty quickly for rare events.

Formula:
Probability of Not Hitting Royal in X Hands:= exp(-1*X/cycle length)

In your case prob=exp(-1.5) ~22%

Video Poker can be a very cruel game. The majority of the time you'll hit a royal in less than 1 cycle, but those long runners can seem endless. There's even about a 1% chance you'll go 4.5 cycles (180,000 hands) before hitting a royal.

Quote: Tortoise

Video Poker can be a very cruel game. The majority of the time you'll hit a royal in less than 1 cycle, but those long runners can seem endless. There's even about a 1% chance you'll go 4.5 cycles (180,000 hands) before hitting a royal.

heh. I'm at about 240k hands now in fpdw. Played 10k hands last week. Deuces x 1. Lost about 700 on 25c. Some really sick days. Burning 100 bill in 350 hands more than once. Sick week. I may rarely play for a while now.