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Thread: Wolverine: probability question

  1. #14
    Don Schlesinger
    Guest

    Don Schlesinger: Answers from Stewart, with my comments

    [Stewart]: The exact probability of at least one run of r consecutive successes in n independent trials, each with success probability p, is given by de Moivre's formula, which I quote from Uspensky's (1937) Intro to Mathematical
    Probability:

    In TeX symbolism,

    $$
    P=1-\sum_{k=0}^{[n/(r+1)]}(-1)^k{n-kr\choose k}(qp^r)^k
    +p^r\sum_{k=0}^{[(n-r)/(r+1)]}(-1)^k{n-r-kr\choose k}(qp^r)^k.
    $$

    [Don]: Wow! I like my way better! :-)

    [Stewart]: In your case, p=2*8*32/(104*103), r=4, and n=5000, so it suffices to consider the first few terms in each sum (the remaining terms are negligible). I get P=0.0245 (rounded). So your approximation was not too
    bad.

    [Don]: Surprisingly close. Doesn't make my way right, but didn't expect to be within one-tenth of the correct answer.

    [Stewart]: In practice, the dealer doesn't necessarily shuffle between hands,

    [Don]: This was foolish of me. Don't know what I was thinking. See below, for what I hope is a much better approximation to the true correct answer.

    [Stewart]: so the true probability will be smaller than 2.45 percent.

    [Don]: Shockingly smaller. Consider that, after three straight naturals, there are only five aces left, and the prob. of getting a snapper is dramatically decreased (by almost half, when one considers the reduced tens, as well).

    [Stewart]: For example, if the dealer has had three straight naturals and is still dealing from the same deck, the fourth natural will be somewhat less likely, because three of the aces will be gone.

    [Don]: MUCH less likely! Suppose I were to deal nothing but four straight hands of two cards each (but with no replacement), to simplify matters. Then the prob. of four straight naturals becomes:

    [(2*8*32)/(104)(103)][(2*7*31)/102)(101)][(2*6*30)/(100)(99)][(2*5*29)/(98)(97)] = 1/1,442,515.

    Compare this result to the 1/191,603 that we got with replacement! The latter is 7.53 times more likely than the former! So, if I now do it my way, I get a new probability of 1 - 0.99654 = 0.00346, or once in every 289 attempts, which, of course, is much rarer than once in 39. What do you get with Uspensky's formula, replacing p with the new 0.000000693?

    Thanks very much Stewart; I appreciate your help.

    Don

  2. #15
    bfbagain
    Guest

    bfbagain: Re: Not so rare

    Don:And, maybe my memory is faulty, and I actually have seen such a streak.

    Considering what I have personally been through of late, I would have to say it's a real stretch that this has not happened to you.

    I can't say for sure if this has happened to me in the last month, but I did have a friend who witnessed my play at a heads up DD game, who was counting the number of dealer draws to 21's, and stopped counting at 39 in a two hour period. Yes, it's rare I would play for two hours, and even more rare at where I was playing, considering the stakes. That said, my losing, and the recent departure - I'm speculating here - of long time personnel probably contributed to my allowed play.

    During this time, a most humbling experience I can tell you, there was no doubt that the dealer had at least three BJ's, and maybe multiple times. I was probably a little too jaded to concern myself with any apparent oddities, as the only oddity was me playing. But a heads up DD, at this place, with no heat, and a place I love to play, well....you don't leave good games, even when you're getting the tar beat out of you.

    Sorry for the verbose answer, as I just got home and was catching up when I saw this thread, but yea Don, I believe you most definitely have witnessed this. :-)

    Myself, if I never witness another 3 1/2 weeks like this, I'll be happy. :-)

    cheers
    bfb

  3. #16
    Don Schlesinger
    Guest

    Don Schlesinger: Rarer than you think, for DD

    > Sorry for the verbose answer, as I just got home and
    > was catching up when I saw this thread, but yea Don, I
    > believe you most definitely have witnessed this. :-)

    I now calculate that you could play 50 hours of DD a year for 60 years and still have a 72% probability of never seeing four consecutive dealer naturals.

    And, for the record, I don't remember ever having seen it happen to me.

    Don


  4. #17
    bfbagain
    Guest

    bfbagain: Clarity is a good thing :-)

    > I now calculate that you could play 50 hours of DD a
    > year for 60 years and still have a 72% probability of
    > never seeing four consecutive dealer naturals.

    > And, for the record, I don't remember ever having seen
    > it happen to me.

    I was referring to three naturals in a row. And for the record, three, four, or more snappers in a row all mean the same thing. You're losing.

    It may be interesting to calculate probabilities, as an intellectual exercise, but dwelling on those things won't help players win.

    The first thing that comes to mind when people ask about what may be perceived as blackjack anomalies, is the perception that there was cheating involved. There may be cheatng, but blackjacks are not the preferred method of cheating dealers, or so I'm told. The reason for the last comment is I probably couldn't tell you who was cheating or who wasn't if I watched with a microscope. However, when I do think I'm seeing extraordinary occurances, I make the dealer very aware that I'm "trying" to focus on their hands as they're dealing. It would take a pretty accomplished cheat to not react to "me" visually focusing, with extreme intensity, on their dealing the cards.

    My conclusions are relatively simple, and that is, no cheating, just bad variance, in almost all cases.

    Bad luck can come in many forms. What most people are not prepared for, are the incredible runs of bad cards that happen. The beans in a jar is always a good example of how streaky blackjack can become, and how small our edge really is.

    The only real thing that pisses me off when I'm in one of these streaks, is the realization that I have to play so many more hours (and so many more hands), just to recover.

    Anyway, I just thought I'd put some perspective to this thread.

    cheers
    bfb

  5. #18
    Karel Janecek
    Guest

    Karel Janecek: Re: Answers from Stewart, with my comments

    > but didn't expect to be within one-tenth of the
    > correct answer.

    Actually, I was expecting that the numbers will be very close. A vague reason is that most trials are indpendent. Only the close neighbours are dependent, but not that much.

    Regards,

    Karel Janecek
    (author of SBA)

  6. #19
    Wolverine
    Guest

    Wolverine: Final answer

    Okay Don, do we have an answer to what the de Moivre's formula indicates without replacement? Your non-independent answer is 1.4 million to 1. In other words, very rare.

  7. #20
    Don Schlesinger
    Guest

    Don Schlesinger: No!

    > Okay Don, do we have an answer to what the de Moivre's
    > formula indicates without replacement?

    No, no further answer from Stewart.

    > Your non-independent answer is 1.4 million to 1. In other
    > words, very rare.

    No, my answer was once in every 289 attempts of 5,000 hands dealt. "Rare"? Not so. The once in 1.4 million was if you try to get four naturals in a row, given one chance at dealing four hands. Here, we have 4,997 consecutive chances. You do understand the difference, right?

    Don

  8. #21
    Wolverine
    Guest

    Wolverine: uh huh

    > No, my answer was once in every 289 attempts of 5,000
    > hands dealt. "Rare"? Not so. The once in 1.4
    > million was if you try to get four naturals in a row,
    > given one chance at dealing four hands. Here, we have
    > 4,997 consecutive chances. You do understand the
    > difference, right?

    Yes, my answer is for 4 straight hands from a DD. Your answer is for four straight naturals dealt from a continuous sample of 5000 hands.

    Sorry, missed the 1 in 289 attempts. That still is a 0.35% chance! That doesn't qualify as rare? You said yourself you don't ever recall having seen this. Nobody else has stepped up to admit seeing it either. bfbagain noted 3 naturals in row. I think we have all seen that. But FOUR?

  9. #22
    Don Schlesinger
    Guest

    Don Schlesinger: No surprises here

    > Yes, my answer is for 4 straight hands from a DD. Your
    > answer is for four straight naturals dealt from a
    > continuous sample of 5000 hands.

    Right. Which was the question you asked originally, no?

    > Sorry, missed the 1 in 289 attempts. That still is a
    > 0.35% chance!

    Right.

    > That doesn't qualify as rare?

    I guess that all depends on how unlikely you require an event to be to qualify as "rare." I'm not sure that there's a mathematical definition of the term. It might be fun to poll the readers and ask them what the probability of an event's occurrring would have to be for them to qualify it as "rare."

    The dictionary says, "infrequently occurring; uncommon." Maybe you're right, after all.

    > You said yourself you don't ever recall having seen this.

    That's not at all surprising. Let's see. 289 attempts at 5,000 hands is 1,445,000 hands, or, at 100 hands per hour, 14,450 hours of play. I've been playing for 30 years, so, I'd have to average 482 hours a year to have accomplished this. I have not averaged anywhere close to that, so it is not at all surprising to me that I have never seen four naturals in a row.

    > Nobody else has stepped up to admit seeing it either.

    We don't have a very large readership. :-)

    > bfbagain noted 3 naturals in row. I think we have all
    > seen that. But FOUR?

    Suppose we have 100 readers. Each attempts to accomplish a 1 in 289 event. Instead, let's say that each tries NOT to have it happen. So, we raise 288/289 to the 100th power, and we get ... drumroll ... 70.7%! If we had 100 readers, and if they all played almost 500 hours a year, there would still be less than a 30% chance that anyone would have witnessed four snappers in a row.

    And, we don't have 100 readers! :-)

    Don

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