another reply...

... I'll break my promise and answer one more time.

>>(1) what is the probability of getting 6 consecutive snapper pushes in 6 rounds of play?

This was not, either from Studog's post, or Eliot's reply, or my reply, ever the question. You introduced it as a red herring, or as a result of your misunderstanding of the original post, or both.

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I didn't introduce it as _anything_ other than to make the point that the probability for 6 consecutive 21 pushes in 6 consecutive hands is quite remote. But 6 pushes in a row, occuring anywhere in a larger sample, is more likely to happen. That was my point. My only point. No intent to obfuscate, or create red herrings, or strawmen, or anything else.

If he had asked "I played 6 rounds last night, got 6 snappers and each time the dealer got one too, does that sound strange?" I'd answer "yes". "too strange".

If he had asked, as I believe he did, "I played N rounds last night, got 6 snappers interspersed with non-21 hands, and each time I got one the dealer pushed. Does that sound strange?" I'd answer "yes, but not as strange as the first case..."

That's all there is to my answer...

Not everyone has a dark ulterior motive when they post...

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>>(2) what is the probability of getting 6 consecutive snapper pushes in 6 million rounds of play?

>>They are _not_ the same...

Thanks for that enlightening revelation. Do you figure Studog played 6 million rounds? Nonetheless, see below.

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No, but I figured he played more than 6. And that changes the answer.

Because now we are into sampling theory, and the larger the number of samples taken, the more unusual some of those samples might appear. Central limit theorem and all that...

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>>That was the direction I interpreted this question to be from...

It's a free country; interpret away. I interpreted that a poor guy sat down to play online BJ, and, in the course of his play (probably a few less than 6 million hands!), he got six snappers, only to be tied each and every time by the dealer. Incredulous, he wondered whether this was remotely possible, if the game is fair.

It isn't. Period.

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"Period". -> _really_ poor math. Ever flip a coin and get 10 consecutive heads or tails? Not often but not "never" either. Would you conclude either (a) the coin is biased or (b) the sample size was too small?

I played a shoe game in vegas, sat down, bought in for $100 and was broke after 22 hands betting $5 flat. 2 pushes. 20 losses. Was the game rigged?

Got a student working on "the blackjack project" in a distributed computing class I teach. He came in and said "this Mersenne twister" looked pretty good until I found that for one case it generated the numbers 1-13 in sequence. That can't possibly be random. I pointed him to the "poker test" and had him read a short paper on the thing. The point being that if he _never_ saw 1-13, his generator is certainly not random. We expect to see 5 consecutive numbers from time to time (AKA straight flush). If we don't something is broken.

So saying that just based on one sample, with 6 snappers spread over multiple hands, where the dealer pushed, that the game is "not fair" "period" is simply poor science. It is an indicator of bias. _not_ a guarantee of bias. takes much more information to conclude something "period"...

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>>Simple, really... No intent to be confrontational, no intent to be anything, other than I felt your answer didn't exactly fit the question as posed.

Why don't we let Studog be the judge of that?

>>Hint:

>>What is the probability of getting 6 consecutive snapper pushes in an infinite number of trials? 1.0

Why don't I go you one better, since talking about an infinite number of trials isn't terribly practical or helpful for our questioner.

Eliot gave the correct methodology for answering the original question, or, at least, for answering the question: "If I sit down to play BJ and for the first six naturals I receive -- however long it takes me to get them -- the dealer pushes me all six times, what are the odds of that occurring?" The correct answer, for the 6-deck game being played, was 0.0456^6 = 0.00000000899, or 1 in 111+ million.

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Where we are disagreeing is here: Out of the N hands he played, there are going to be M groups of 6 consecutive snappers with intervening hands between each snapper (or not). So the more hands he plays, the more likely it is to see the above happen. For example, playing just 7 rounds, the probability for that happening is now 2x the above, since the first 6 or the last 6 could be consecutive pushed snappers. This keeps adding up as you add more hands (simple combinatorics).

That was my point. Yes it is unlikely. But no it is not exactly 1 in 100+M hands. In BJ, it is always possible that this "session" was that one in 100 million session. Where statistics says "OK, it is time..."

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Now, suppose you say, well that's for just the first time the guy plays. Let's do this over and over again, making lots of attempts (but not infinite!) to get this to happen. How about a MILLION??!!

So, we try a million times. Probability that the dealer would push us, 6 for 6? 1-(1-0.00000000899)^1,000,000 = 0.00895, or once in 111 tries (less than one percent likelihood)!

How many times would we have to do this until the chance that we would encounter the 6 dealer pushers would be, roughly, 50%? Take a guess before you look. Answer ... 75 million attempts!! (Close enough to "infinity" for you, considering how long our mere mortal lives last?)

To Studog: Bottom line -- You are free to interpret the advice you received here any way you choose. Here is mine: Unequivocally, without the slightest doubt whatsoever, you were cheated. Avoid this game like the plague.

To SSR: Play the game to your heart's content; doesn't seem like such a rare event to me! :-)

Don

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I don't play online period, because being a computer scientist I understand just how easy it is to rig things. And I have talked to some of the developers of these casino systems that talk about "dialing in a house edge" which obviously means that the cards are not exactly "random" but dealt to produce some specific hold.

But, by the same token, this one sample is hardly enough to say "this site cheats, absolutely". Insufficient data. Many similar samples would support that conclusion better. Even better would be to do what Shackleford proposes, capturing each card dealt. That can _easily_ be confirmed to be random or non-random, with a high degree of confidence, rather than just remembering one oddball happening out of an unknown number of rounds played.

Otherwise I could disprove lots of our BJ lore.

Two weeks ago, I played and saw several unusual things. I had two consecutive snappers that pushed (two consecutive rounds.) Ought to happen once every 160K rounds or so correct? I had 6 different chances to double 9, 10 or 11 vs dealer 6 and lost _every_ one of them. Obviously that proves BS is broken.

Or is it too small a sample.

I think of this this way: 6 snapper pushes is strange. Losing N consecutive high-probability doubles is strange. Doubling on 11 and getting an A is strange. If you enumerate all the strange things that can happen, then in a given session the probability of one strange thing happening becomes quite high.

At least in the games I play it happens a lot. Personally I learned years ago that strange things _do_ happen. The more "strange things" we can recognize, the more we will actually see also...

So you conclude they are absolutely cheating. I conclude they deserve watching. Which is right? Perhaps time will tell. But for anyone really expecting an honest internet BJ game, I have some real estate for sale in ...