Significance.
Alright, here goes nothing about "long-term" math since someone said nobody did. We need to be very explicit about the question being asked to talk about variance or expectation because I'm sure it is confusing!
Some Questions:
1) How many royals can I expect from 20 million hands and with what variance?
2) What is the expectation and variance of my "average payoff" over all 20 million hands?
3) What is the expectation of the amount of money I'll end with after 20 million hands? What is the variance?
Okay so video poker, assume optimal card holding, it's just an empirical distribution (specific probabilities for each payout). You can use standard formulas to calculate the mean and variance. For arguments sake, assume it is a 100% EV game ($0 goes to you) and we'll take the variance of 40 above. But just know that you can only hit certain values if you played let's say 5 games ($-25 to $5royals).
Expectations and Variances:
1) you expect to get one every 40,000 hands I think. Your distribution is binomial p=1/40k and n=20 million. Resulting expectation is np= 500 and variance is np(1-p) = 500, so standard deviation is 22.
2) This is where your square root of n comes in. The average of 20 million independent games will have expectation $0 and standard error sqrt(40)/sqrt(20 million) which is like 1.5 in a 1000 --- in other words super small. But, nobody cares much about this sampling distribution?
3) This is the question people care about usually because it's your money. Unfortunately, this is more up and down. Although the expectation is still $0, the variances usually sum, so you're talking about 40 times 20 million which results in the original poster's figure of 28,284 after the square root.
Statistical significance:
1) As np is large, you can use the normal distribution z-values to approximate binomial. So, if you got under 500-22(2) or over 500+22(2) royals, then okay, "you're significant" which means you hit the unlucky 2.5% or lucky 2.5%. Given this is gambling and the machine should have constant probabilities, I guess that would be luck.
2) I'm not sure if anyone is concerned about this? But the analysis is straight forward - take average of all your hand payouts and compare it to Normal (0, std. err. = 1.5/1000)
3) Same thing as the previous but it's Normal (0, s.d. = 28284). So if you're down 28284(2)=$56k or up that amount, you're statistically "unlucky" or "lucky."
Bottom line: Sums much more variant than averages. The significance depends on the "long term" or "short term" and that was modeled with the "n" which then contributes to the variance. Long term didn't matter much for question #3, but it did for question #1. For question #2, it mattered very little.
So, yes, I think this is long-term enough. But, of course we ignored all sorts of things like cash back and the exact distribution of video poker, although only independence of outcomes of sequential games was usually sufficient. Although the answer to #3 may seem not bad for that amount of money, you're going to get tanked if that EV goes below 0 by the number of times you play. The cash-back is a constant, so it just adds to the expectations in # 2 (per hand) and #3 (per hand times number of hands), it changes the location of the distribution. It is irrelevant to the first question obviously.
The distribution for Question 3 is really why the casinos win against most people. They can stand the swings in variance and you can't. So play wisely or play for fun...
All else equal, I would say the 200k units ahead surely is statistically significant (a p-value very close to 0 for #3) to a point where the game itself could be in suspicious - - - and this suspicion is attributed mostly to the smart plays of positive EV and cash-back, unless there was a faulty machine.
---In vpfree@yahoogroups.com, <007@...> wrote:
I hope I've done this right, since I was never very sure of myself
regarding how to use probability. I hope I'll be corrected if not:
Assume Jean and Brad play 800 hours per year. Assume they play 1000
hands per hour. Assume they always play 5 coin single line $1
machines with a variance of 40. They've played 20 million hands.
Their standard deviation is the square root of the number of hands
times the variance, or 28,284. They're 200,000 units, or 7 standard
deviations, ahead. If that isn't statistically significant, what is?
RWS wrote:
>Well congratulations on your results. But that information doesn't address my original observation that those who play VP advantage really don't understand that they are not playing in the long term as far as statistical significance. It's very likely that in your 25 years of play, your number of hands doesn't approach the long term either.
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>________________________________
> From: Queen of Comps <queenofcomps@...>
>To: vpFREE@yahoogroups.com
>Sent: Saturday, November 9, 2013 4:54 PM
>Subject: Re: [vpFREE] RE: Today I was dealt a Sequential Royal Flush!
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>Playing over 25 years, only on positive VP plays (which include all cash
>extras, i.e., free play but not including comps), we have won an average of 40K
>per year.
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>------------------------------------------
>Jean
>$¢ott, Frugal Gambler
>http://queenofcomps.com/
>You can read my blog
>at
>http://jscott.lvablog.com/
>
>From: RWS
>Sent: Saturday, November 09, 2013 2:06 PM
>To: vpFREE@yahoogroups.com
>Subject: Re: [vpFREE] RE: Today I was dealt a Sequential Royal
>Flush!
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>How
>is it that anyone feels their playtime at video poker even approaches long term
>math? (millions of trials) I have yet to see anyone explain
>that on this site.
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>________________________________
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