> So, in your first example, heads comes up 75
> times out of 100 and leads to a windfall of
> profits. That is 1.56 standard deviations
> (which equals 16) from the mean of 50. An
> event that is likely to occur. The *next*
> trial of 100 coin flips is not more likely
> to come up 25 heads to balance it out, or
> any other number. However, over the long
> haul, in order for the classic bell curve to
> balance out, IT IS MORE LIKELY that a less
> than 50 heads result will occur in the
> future! If it does not, then the mean over
> the long haul of these 100 flip trials will
> not equal 50. When that less than 50 heads
> trial will occur, nobody knows. But it will
> happen. You can't keep getting 100 event
> trials that equal 75 heads either, or again,
> the mean will not come out to 50 over the
> long haul. If we differ on this basic
> premise, then I can certainly see where we
> have a fundamental difference of opinion
> regarding statistics and how they are
> interpreted. If someone can point to a
> fundamental error in this logic, I would
> love to see an explanation and learn from
> that explanation.

There is no "difference of opinion" you simply have not yet grasped the concept of really large numbers. Suppose that, after our initial 100 flips, the remainder of the million flips was exactly half heads and half tails. Guess what? We're still the same. The 75 out of 100 that we started with has now become statistically insignificant. Fred Renzey expressed it well in Blackjack Bluebook II: "Lop-sided results are not corrected; they just fade into the past." The ever-accumulating number of normal outcomes has diluted the impact of those first 100 flips until they've become virtually meaningless. This is the key: there is no "equal and opposite" reaction.

> Again, I agree that the odds are exactly the
> same, we have an edge on every coin flip!
> However, many 100 coin trials have occurred
> without us participating in the outcome. We
> have no way of knowing what those trials
> have been. Will those trials influence the
> next session of 100 flips we choose to wager
> on? The coin is still 50-50 and the
> statistics say that we can get a result from
> 2 to 98 heads and it would be within the
> statistical expectation. However, over the
> long haul, if the trials which have been
> going on without our knowledge have
> accumulated on one side of the mean (50)
> than another, than the long haul pull toward
> the mean of those 100 flip trials may exert
> its power on the next 100 flip trial. Since
> we weren't there to participate in the
> winning (or losing) streak that has occurred
> in our absence, we are at the mercy of the
> long haul average influencing the next 100
> flips. No?

NO! These are independent trials. Past results have absolutely no effect on future outcomes. Once an event has already occurred, the probablity of it happening becomes 100% because it has already happened. See above.

> This is exactly why I am asking the short
> term question about how to approach playing
> a given blackjack session. At some point in
> the 100 flip trial, you are up more than you
> would expect. Is it a good strategy to
> pocket your forutnate winnings early and
> await another trial in the future or
> continue to bet on the remaining coin flips?
> In other words, should you pocket your
> excessive winnings before the trial is over?

Again, this only works if you plan on never again playing in your lifetime. Otherwise, it makes absolutely no difference (from a mathematic standpoint) whether you continue playing, play an hour later, the next day or a week later. However, the only way to reach the long run is to put in the hours - you have no EV when you are not playing.

> Similarly, if you are down early (lose the
> first 5 flips, or something like that)
> should you take your lumps and walk away
> before the trial ends, again, to await
> another trial in the future?

What causes the trial to end? It's all one trial. The breaks are meaningless, except that you are wasting valuable time.

> I can see that cutting your losses early
> makes little sense in this scenario. It is a
> can't miss proposition when you get a 50%
> premium over the true odds. Play out the 100
> flip session and accept your fate.

> But when your possible advantage is only 1%,
> and you may never get back to even, let
> alone a winning position, due to standard
> deviation: is cutting your losses to
> minimize them a prudent strategy to allow
> your winning session a chance of recouping
> those losses?

The fact that your edge is smaller and variance is larger is all the more reason to get in as many hours as possible. It simply means that it will take you longer to get to the long run.

> Additionally, blackjack isn't divided into
> even trials like we have laid out in the
> coin example. Each deck presents a different
> "value" to the advantage player to
> win. Some are big losers, others big
> winners, most are in between. You have to
> "start" and "stop" a
> blackjack session. Perhaps the best way to
> describe it is that you are playing the last
> 25 flips of one trial, and the entire next
> 100 flip trial, and then 75 flips in the 3rd
> trial. You have created your own 100 flip
> trial times two. Perhaps the first
> "original" 100 flip trial that you
> only caught the last 25 of, was good early,
> and bad late. You come out a net loser for
> that 25 trials. The middle 100 and you play
> with a "formal" 100 flip trial, is
> dead even, 50-50. The last 75 you play
> breaks even as well. You wind up a net lose
> thanks to the first 25 flips you jumped
> into.

No. You are assuming that the outcome of every 100 flips is somehow predestined. It isn't. Streaks exist only in hindsight. Any streak can end (and another streak begin) at any given time. Streaks can continue through breaks and folllow you from one table (or casino) to another. You have absolutely no way of knowing, and certainly no way of controlling this, so you play for the long run.

> I think I am beginning to see the light. I
> have to go get dinner, but the light is
> beginning to go on. Eureka! Thanks. I'll
> keep working on this. Cool.

I certainly hope so. :-)