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Thread: Wolverine: Play all spread

  1. #14
    Wolverine
    Guest

    Wolverine: Then where are the past posts?

    > There is nothing new or unique in your post.
    > We have been down this road 1000's of times
    > before. You either accept the math or you
    > don't. The math doesn't care either way.
    > Neither do we.

    I'm sorry to hear that you don't care. And if you have been down this road 1000 times in the past, where are the archived explanations for us "newcomers" to learn from. I am simply trying to reconcile my life experiences with the education you espouse here. To dismiss my discussion and subsequent education as you have is very disappointing.

    I would hope that someone is able to shed some light on what my experiences have been. Thank you.

  2. #15
    bfbagain
    Guest

    bfbagain: OK, I'll bite...somewhat, but first,

    let's start by you telling us what you have read, and what you haven't, but think it's something that you think you should read.

    There's nothing wrong with asking questions, but the truth is, the game of blackjack now has had, what, somewhere in the neighborhood of a few trillion sims run, hundreds of mathematicians conducting research, casinos bankrolling the games with the current rules for, 50 years give or take a few.

    Much of the information that would seem to satisfy you is already in the public domain, and as I previously stated, for many years. But you're right, you have asked, and someone should help answer.

    So I'll start with this:

    The only money management that works is understanding and utilizing kelly. The proper sizing of your bets with respect to your current bankroll, the game your playing, and the edge you currently have to employ optimal bet sizing is all that counts.

    You cannot control variance. In other words, luck, as that's what variance is. The money management, a.k.a., kelly criterion, allows you to withstand the negative fluctuations while you build your bankroll. That's it. Maybe a little oversimplified, but that's it in a nutshell, and anything else is, as Parker aptly stated, voodoo nonsense. You may not want to hear that, and that's fine. I'm sure there are patients that don't want to be told they should quit smoking or they'll die, but that doesn't mean the Doctor's advice is any less accurate.

    As I often would tell my people (once upon a time), you can talk all day long about the sun, but it still is going to rise in the east, and nothing they could say is going to change that.

    It's the same with attempting to control session wins and losses, via money management. Attempts to use money management is so people who are playing a negative eV game, e.g., craps, won't lose their shirt. People like us, who are playing WITH an advantage, want to keep playing as long as possible, as we're in effect, doing the same thing as the casinos, playing with an edge. They want as many people playing with negative expectation as they can get in the door. Do they sometimes lose to those people, of course. Do you think if they had a losing month or two that they should close down? Or stop the tables at some preset point? Their shareholders would go nuts. See where I'm going here? I hope so.

    Now, am I going to advise AGAINST someone who has been losing steadily, to NOT stop with a win, if it means that will help their personal confidence level? Of course not. Confidence plays as big a role as many things. However, as I have often cautioned people at the tables, my attitude can get remarkably better when I start hitting my double downs with max bets out, as well as the occasional blackjack or three with max bets.

    But I have no control over that. You only lose when you QUIT, PERIOD!

    The only time you leave a great game, is 1) It no longer is a great game, and or 2) You don't have enough money on hand to cover all possible bets, i.e., not enough to cover splits, resplits, and/or doubles.

    cheers
    bfb

  3. #16
    Wolverine
    Guest

    Wolverine: That wasn't so hard, was it

    thanks bfb, that is much nicer and clearer. And I play a winning game of BJ (now--finally! thanks to what I have learned here and practice, practice, practice), but I still have winning and losing sessions. That is what is prompting the questions. I look back at sessions and wonder why things happened the way they did. Call it an M&M (morbidity and mortality) Report on the session. If I'm up X amount of money, and the count was positive, why did I keep playing and wind up a loser? Was there something I could have done differently to affect that losing outcome? Same thing with winning sessions. What went right and what allowed me to walk away a winner? Why didn't I keep playing until I was back to even or a loser? I must have walked away for a reason, but no answer has been forthcoming from this site. I look at my investing background of dealing with variance for answers, because to me BJ and investing are VERY much related. Obviously, others do not share that view at all.

    Thank you for the understanding. And I have read through the "best of the masters" and the "distinguished posts" in DD before I ever started this thread and thought process I've pursued here. Session bankrolling and variance as I am discussing it are not archived there (even though they have been discussed and dismissed a thousand times!). I have read BJA3 through once. Have read parts of it twice. I've made my own BS cards for personal use from the appendix (not with magic marker, on a computer, thank you :-) ) I have read other books on BJ too. I enjoy math and statistics. But individual session variance is an interesting subject to me (I call it money management and psychology of BJ, others disagree there are such things). I am not a pro. I have a day job which I love very much and am very successful at. I am a college graduate. I enjoy a good debate and thought provoking dialogue. Why is this such a tough topic to talk about without just dismissing it as voodoo? I talk of populations and samples, I get told it doesn't work that way. That is the way statistics work! The very thing that you all talk about is variance or S.D. in a session, and point to the fact that it is uncontrollable. I am simply begging to differ and ask the tough question: If you have the chance to walk away a winner from a session and you don't, why? If you never had the chance to be a winner and struggle to get back to even, but fail to and lose your bankroll, why? Why walk away from a session after a given period of time (45 minutes to an hour like BJA3 and other posters keep talking about) so as to not give away that you are an AP? These are legitimate questions and deserve legitimate debate! Not dismissals as voodoo session superstitions. Show me the error in my statistical logic about sessions and I will listen with open ears! When I ask about 12 vs 4 and 12 vs 6, Don points out the mathematics of the sims. And I thank him for it and continue my learning journey.

    Maybe I'm missing something here, and I certainly hope to learn from it. But give me reason to learn, that is all I ask. Maybe one of my fears is from all that I have read here that much of what is published about BJ is false. That is why I have enjoyed this site so much is the brainpower and the ability to discuss things, logically and with a bent toward the mathematics of BJ. I know that I have to purchase Norm's software and begin the journey toward using sims to evaluate the game of blackjack myself.

    Thank you for listening....

  4. #17
    Wolverine
    Guest

    Wolverine: Sims

    bfb,
    You talk about a few trillion sims run on BJ. Yes, yes, yes. Sims are great for the long run, and help us tremendously in understanding BS, advantage play, the indices needed for each game, the Ill-18, the Catch-22, etc... Each of those remarkable discoveries were made with the help of the massive simulations achieved by the growth of computing power. No debate there whatsoever.

    A trillion sims don't mean diddly in a single session other than to give you the "rules" to play by: if the game has X number of decks, can only double on 10 and 11, no double after a split, etc... The sims help to create the "best playing strategies" given that scenario for the long haul. Agreed.

    But, if you are getting your butt kicked, are you willing to lose your entire (supposedly adequate) bankroll in a single session just to prove that things are going to get better with the next hand because the game is good?

    And if you read my post above, why is one session (starting and stopping) different than a buy and sell in the stock market? Both have positive long term trends, but if you walk away a loser over and over again in either, you will be a loser over the long haul too! I'm trying to examine that specific issue. In investing, there is asset allocation and stop loss rules to prevent the destruction of a portfolio (can anyone say Enron 401(k)?). Everyone seems to think there is no way to stop the bleeding in BJ, because we are all playing for the long haul. I am simply asking for a logical explanation (not necessarily a mathematical simulation) for why a sample of BJ (a session) is different statistically from the stock market and how we as individuals influence the outcome of that session independent of the BJ playing skills (counting, bet ramping/ranging).

    And yes, I'm looking for the sun to come up in the East tomorrow. :-) And the next day, and the next day, and... :-)

  5. #18
    bfbagain
    Guest

    bfbagain: More on session objectives

    There are many nuances that go into a battle plan, i.e., session strategies, to maximize gains. Some of these relate to exposure. Some relate to heat. Some relate to longevity. Not all the previous terms are the same, as they often relate to context.

    For example, playing for no longer than 45 minutes, although normally a hard and fast rule for a high stakes player, is hardly an issue for a red chip recreational player, on a four day trip to Vegas. Likewise heat, and how it relates to longevity.

    However, having a mandatory "time-out" requirement, if after playing a certain length of time or losing a pre-determined amount, e.g., a session bankroll, is perfectly acceptable. This is NOT money management. Some teams incorporate that also.

    Another example would be a "win limit." If playing anonymously, and you're approaching 10K, you may choose to stop playing so as not having to have a CTR filed, where your identity would be disclosed. Again, not money management, but part of an overall strategy.

    Comps also play a role. Shift changes. And essentially, the list goes on and on, for reasons that you stay and play or get up and get away.

    So be careful what you read, and to what audience the post was for. If you're not sure, ask.

    cheers
    bfb

  6. #19
    Parker
    Guest

    Parker: Re: Then where are the past posts?

    > I'm sorry to hear that you don't care. And
    > if you have been down this road 1000 times
    > in the past, where are the archived
    > explanations for us "newcomers" to
    > learn from. I am simply trying to reconcile
    > my life experiences with the education you
    > espouse here. To dismiss my discussion and
    > subsequent education as you have is very
    > disappointing.

    If my responses seem dismissive, it is because the overall tone of your posts make it difficult to discern whether you are genuinely interested in learning something, or merely looking for an argument, aka trolling. If the former, please accept my apologies. If the latter, I really don't have the time.

    One of the things that makes advantage play difficult is that until you have been playing the game for many years, it is impossible to draw any meaningful conclusions from "life experiences." Your sample size is too small and variance is too large.

    You talk of ending a session prematurely in order to preserve a large session win. This only works if you intend to never again play in your lifetime. It makes no difference whether you start again in 5 minutes or 5 weeks, the odds are exactly the same regarding your subsequent success or failure.

    Besides, how do you know that you will give it all back if you continue to play? Answer: You don't. It is just as likely that your streak may continue, further increasing your winnings.

    We accept some constraints as far as limiting the length of our sessions, but these have nothing to do with the mathematics of the game. We keep our sessions short so we may continue to be allowed to play, and because we are human and eventually tire. From a mathematic point of view, the more hours we play, the more quickly we reach the long run, and the more quickly our results fall in line with our expectation.

    We start every shoe (or pack, or deck) with the same expectation. Sure, streaks happen, but they can only be predicted in hindsight (how's that for a twisted metaphor). Ending a session has no effect on a streak, good or bad. Why is the streak any less likely to continue after you take a break, switch tables, or even quit for a few weeks? Answer: It isn't.

    As for archived posts, you probably won't find them on Don's Domain, since this forum is relatively new. Try searching for "long run" on my main blackjack page.

    You might also consider reading Blackjack Bluebook II, by Fred Renzey. This book contains about the best explanation of probability and the "long run" in plain English that I have seen anywhere.


  7. #20
    Wolverine
    Guest

    Wolverine: Thank you

    I am sorry if it seems I am looking for an argument, I am merely looking for some sunlight on a subject I find fascinating. I have homework to do, and I will look into the posts you are talking about and the Renzy book as well. I know I need software as well to work on sims, indexes, etc...

    But your post brings me right back to my initial point I started to harp on: you limit your wins due to time and or amount won (for whatever reason). Why wouldn't you try to limit your losses as well? And I will not fire back about sessions being small samples of the statistical population of blackjack until I do more homework. Let's just say, nothing said here has given me EVIDENCE that my statistical thinking of each session as a sample of the population is incorrect. You just keep pointing to the long haul, and asking me to believe that each session will regress to the long haul mean on faith. I have a wholly different interpretation of session statistics. But I will hold my fire until I find someone who wants to discuss them, AND I have done the homework you have suggested.

    I have been playing BJ using BS since at least 1990, so 14 years isn't nearly a lifetime, but it is longer than many. I have only been interested in counting for the last 4 years or so. Don, on this site and his book, has helped with my bet ranging and understanding what I am trying to do with my wagers. I have about a dozen books in my BJ library. Some are complete crap and I saw through them quickly. Others make sense and have added valuable insight into playing strategies. Thank you for your patience and discussion.

  8. #21
    bfbagain
    Guest

    bfbagain: Who says you don't?

    Limit your losses that is. What seems to be missed, or maybe the concept of 'talking past people" is in play here, is still your persistence that there is a money management type of strategy that can be successfully used.

    You have said, something to the effect, "if you lose 12 hands in a row, even in large positive counts, why wouldn't you stop?", or something similar. The reason is you don't stop when you have an advantage, as that's what you're playing for. Again, the money management issue is only relevant if these were independant trials, with negative expectation.

    Maybe I'm missing something here, but is that concept difficult? It's not about sample size, it's about what is mathematically correct.

    Is it possible that counters, even when playing mathematically perfect, can lose? YES!

    Is it possible that a counter will NEVER see the day that the LONG RUN exists in a positive way? YES.

    If the fear that one day you will never get there, i.e., that your "edge" will never manifest itself, then this is not the game for you. And despite your reticence to simply "take people's word for it", of which no one is asking that of you, you will start to dangerously take on the similarity of an internet, a.k.a., blackjack troll.

    One last time. Limiting your session wins and losses have nothing to do with money management. They are strategically based. Kelly betting is what constitutes money management.

    Sorry, but this is why respected people try not to get engaged in this type of discussion. I said I'd bite, fully knowing, that this is exactly the type of conversation that has taken on lives of their own all over the internet, for years, on many web sites, including the old newsgroups, e.g., rec.gambling.bj.

    So, if you continue to, and believe me, the information is out there -- and yes, it'd be highly advantageous to you to buy the necessary software, IMMEDIATELY -- so you can confirm to yourself what the majority of people have already learned is true, you'd be doing yourself a tremendous favor.

    Good Luck to you
    cheers
    bfb

  9. #22
    Wolverine
    Guest

    Wolverine: Re: Who says you don't?

    bfb, I appreciate your input. I see what you mean by the long run. I have said that already. And if I could possibly play 20M hands of blackjack over a lifetime, then the long haul should manifest itself. [Simple aside here, skip if you wish: If I play 150 hands an hour (straight up vs. the dealer) and play for 10 hours over a weekend, that is 1500 hands per weekend trip. If I do that 4 times a year, I'll play 6000 hands a year. It will take 20 years to get to 120,000 hands at that pace. Is that enough trials to get to the long haul? I don't know, I have to work with a simulator to discover that for myself, which I will do!]

    My question(s) are not about the long haul, they are about the short haul, aka "a session."

    > You have said, something to the effect,
    > "if you lose 12 hands in a row, even in
    > large positive counts, why wouldn't you
    > stop?", or something similar. The
    > reason is you don't stop when you have an
    > advantage, as that's what you're playing
    > for. Again, the money management issue is
    > only relevant if these were independant
    > trials, with negative expectation.

    I said something similar to that, and I'm sorry if it was confused with what you are trying to point out. I summarize what you are saying as: If you have been counting, and the count is positive (wildly positive even), that is exactly what you are looking for. Counters would LOVE to plunk down a big, fat bet given the positive count (aka, advantage) the count indicates. And if you were walking by and *knew* that the count was wildly positive, you WOULD plunk down a big fat bet. Did I capture what the concept is? I hope so. Because that is something I understand. You put more money on the table when you have the advantage.

    > Maybe I'm missing something here, but is
    > that concept difficult? It's not about
    > sample size, it's about what is
    > mathematically correct.

    If I summarized your concept correctly above, then NO, that concept is not difficult. You want to put money out on the table when you have the advantage any time you can.

    > Is it possible that counters, even when
    > playing mathematically perfect, can lose?
    > YES!
    > Is it possible that a counter will NEVER see
    > the day that the LONG RUN exists in a
    > positive way? YES.

    I need to do some sims and research to understand what the expected win rate and the standard deviations are in order to formulate any thoughts on why that outcome could happen.

    > you will start to dangerously take on
    > the similarity of an internet, a.k.a.,
    > blackjack troll.

    What is that. It doesn't sound good though.

    > One last time. Limiting your session wins
    > and losses have nothing to do with money
    > management. They are strategically based.
    > Kelly betting is what constitutes money
    > management.

    Yes, that is what I am driving at. Strategy based wins and losses. This site uses different terms than my past BJ education/reading. I consider money management to be anything that deals with: overall bankroll, session bankroll, optimal wagering, actual wagering, camoflauge (sp?), comps, limits on winning, limits on losing, when to leave, when to sit down, etc...

    Many limits on sessions have been pointed out: time, hunger, being tired, facility breaks, being tabbed a counter, winning too much, etc... I'm merely asking the converse question as well, when should you limit your losses. So far, everyone says that limiting your losses isn't necessary. But *I* haven't uncovered the piece of information that confirms that playing strategy. I would like to! I know, I know, I still "don't get it." That is why I will do my homework and try to discover an answer that will satisfy my curiosity about this. I just need to stop looking at this and do my blackjack homework. We keep getting off the track I am trying to work on regarding short haul results.

    Thanks again for your time and patience.


  10. #23
    Parker
    Guest

    Parker: Back to basics

    Perhaps the concept will become clear if we reduce it to it's most basic form: A coin toss. Assume that we are betting $1 per toss on a simple coin flip. It is an unbiased coin, the odds are exactly even that it will come up heads or tails on any toss. However, the person offering us this game is math-impaired and has agreed to pay us $1.50 every time the toss comes up heads. If it comes up tails, we merely lose our original bet. Obviously, we have an edge.

    So, suppose we start playing, and after 100 tosses, the coin has come up heads 75 times. We expected it to come up heads 50 times, but we have experienced some short-term variance -- positive variance in this case. We have won well in excess of our expectation. Does this mean that we should now quit playing? Of course not. We have exactly the same edge as we did before we started. We still have the same positive expectation.

    Since we are ahead of expectation, does this mean that the coin is somehow now more likely to come up tails? Can we now expect to lose 75 of the next 100 tosses so that we will return to expectation? Of course not. The odds of each individual toss are still exactly 50-50. The next 100 tosses (or any 100 tosses) are no more or less likely to end up tails 75 times (or any other combination) than our original 100 tosses.

    Will the numbers somehow change if we stop playing for 10 minutes, a week, or a month? Of course not. The odds remain exactly the same.

    Contrariwise, suppose we only win 25 of our first 100 tosses. Will we accomplish anything if we stop playing now? Of course not. The only way to recoup our losses is to continue playing. The odds will be exactly the same, now or later. Every toss is still a 50-50 proposition, and we are still playing with the same edge as when we started.

    Since we have a positive expectation on every toss, our strategy should be to play as many tosses as possible. The only "money management" we need is to start with enough cash on hand so that an early negative swing will not wipe us out. We cannot predict the outcome of the next toss, or the next 10, or even 100 tosses, but we know that if we hang around for a million tosses, we will make pretty close to a quarter-million bucks. Furthermore, it makes no difference at all if we make those million tosses all at once, or in groups of 1000 or 10,000 or any other combination.

    See how simple this is? Blackjack is no different. Each shuffle is like a new coin toss.

    We're not asking you to accept anything "on faith." The math is the same, whether you believe it or not. You can choose not to believe in the law of gravity, if you wish.

    But don't go jumping off any tall buildings. :-)

  11. #24
    Wolverine
    Guest

    Wolverine: Re: More on session objectives

    > There are many nuances that go into a battle
    > plan, i.e., session strategies, to maximize
    > gains.

    > However, having a mandatory
    > "time-out" requirement, if after
    > playing a certain length of time or losing a
    > pre-determined amount, e.g., a session
    > bankroll, is perfectly acceptable. This is
    > NOT money management. Some teams incorporate
    > that also.

    This is exactly what I am looking for. Losing a certain amount in a session! What is an amount that is acceptable? 5% of bankroll? 10% of bankroll? 20% of bankroll? Why that amount per session?

    > Another example would be a "win
    > limit." If playing anonymously, and
    > you're approaching 10K, you may choose to
    > stop playing so as not having to have a CTR
    > filed, where your identity would be
    > disclosed. Again, not money management, but
    > part of an overall strategy.

    Why is a win and loss limit/strategy NOT money managment [MM]? I consider MM to be a lot of things, including actual wagering, optimal wagering, session wins and losses, etc... Are you referring to MM as only wagering? For example, a "trick" such as a Martingale technique? Yes, I know that doesn't work. NO need to educate me there.

    Thanks again, bfb for your time and efforts.

  12. #25
    Wolverine
    Guest

    Wolverine: Re: Back to basics

    Thank you Parker, I will take your advice and look into this example to more clearly understand your thoughts about blackjack expectations and the long haul. It is very well worded: clear and concise. I would think an archive of this post would be helpful to point "newbies" to when someone asks for the 1001st time about what I am asking about.

    That way you can save yourself from becoming frustrated with them. ;-) Thanks for your continued help.

  13. #26
    Wolverine
    Guest

    Wolverine: Re: Back to basics

    > Since we are ahead of expectation, does this
    > mean that the coin is somehow now more
    > likely to come up tails? Can we now expect
    > to lose 75 of the next 100 tosses so that we
    > will return to expectation? Of course not.
    > The odds of each individual toss are still
    > exactly 50-50. The next 100 tosses (or any
    > 100 tosses) are no more or less likely to
    > end up tails 75 times (or any other
    > combination) than our original 100 tosses.

    Okay, I've looked over the post and here is my statistical interpretation, and this is where we keep getting hung up. I look at the example you have given and break it down into an overall expectation of 50 per 100 flips will be heads (by definition, it is a fair coin) (the population mean will be 50%) with a s.d. of roughly 16. 50 - (3 x 16) = 2 and 50 + (3 x 16) = 98. That way 98% of the "each 100 flips" trials will fall within that 3 s.d. range per statistical analysis.

    You are 100% correct that *each* coin flip is an individual event with 50-50 odds of coming up heads or tails.

    By definition, the overall population average of ALL flips will be 50% each. a.k.a., the long haul. How many flips, 1 M, 20M, 100M, etc... Big number.

    Each trial of 100 coin flips however, is a different animal. The number of heads (or tails, doesn't matter which one you count) in each 100 event trial will average 50, but each event can range from roughly 2 up to 98 with the numbers closer to 50 being represented more often thanks to the classic standard bell curve. This is basic, Statistics 101 stuff, but thought I would outline it for anyone that hasn't been in that class.

    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.

    > Will the numbers somehow change if we stop
    > playing for 10 minutes, a week, or a month?
    > Of course not. The odds remain exactly the
    > same.

    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?

    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? 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?

    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?

    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.

    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.

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