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Thread: "Probability of Being Ahead by a Certain Amount" Question

  1. #1


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    "Probability of Being Ahead by a Certain Amount" Question

    Hello all,

    I'd like to calculate the probability of having a +5.00% EV after flat betting 5,000 hands while playing a -.45% player advantage game of double-deck blackjack. Is it less than 1%? Can someone please show the math (z-statistic, etc.)?

    I used Norm's Outcome Calculator https://www.qfit.com/calcrisk.htm
    to answer this question. But his calculator requires entering monetary parameters (such as bet size and desired financial outcome) and not just the above terms, and I am not sure I entered the correct numbers.

    Again, using the above numbers:

    When flat betting $10 for 5,000 hands (by the way, can "hands" be said to be exactly equivalent to "rounds?") and ending up with a player EV of +5.00% while playing a -.45% player advantage game of double-deck blackjack, does this mean the player ends up with $2,500 ($10 X 5,000 hands X 5.00%)?

    If this is true, again, is the probability of this occuring less than 1%?

  2. #2


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    Quote Originally Posted by Overkill View Post
    Hello all,

    I'd like to calculate the probability of having a +5.00% EV after flat betting 5,000 hands while playing a -.45% player advantage game of double-deck blackjack. Is it less than 1%? Can someone please show the math (z-statistic, etc.)?

    I used Norm's Outcome Calculator https://www.qfit.com/calcrisk.htm
    to answer this question. But his calculator requires entering monetary parameters (such as bet size and desired financial outcome) and not just the above terms, and I am not sure I entered the correct numbers.

    Again, using the above numbers:

    When flat betting $10 for 5,000 hands (by the way, can "hands" be said to be exactly equivalent to "rounds?") and ending up with a player EV of +5.00% while playing a -.45% player advantage game of double-deck blackjack, does this mean the player ends up with $2,500 ($10 X 5,000 hands X 5.00%)?

    If this is true, again, is the probability of this occuring less than 1%?
    Overkill,

    Ok, I'll give it a shot!

    First, if you play 5000 rounds at $10/round on a -0.45% game, what is the expected value, or EV? That's simply the "handle" (total amount of initial bets) times the edge, so

    EV = (# of rounds)*(flat bet per round)*(edge) = 5000*$10*(-0.45%) = -$225

    So, if you were to repeat this 5000-round experiment many, Many, MANY times, on average you should expect to lose $225 each time. However, the chance of any one 5000-round trial ending at precisely -$225 is very small.

    Now the missing piece of information in your problem is the standard deviation (or, alternatively, the variance) per round for flat-betting this hypothetical DD game. Typical values are in the 1.1 to 1.2 range: the actual value depends on the house rules. I'll use 1.14 units in this post.

    So, if the average result is -$225, what range of results should you expect? Here's where we need the standard deviation, or SD. The SD is calculated by multiplying the SD per round (in units) times the flat bet per round times the square root of the number of rounds, so:

    SD = (1.14 units)*($10)*sqrt(5000) = $806.10

    Now if your results follow a typical bell curve, statistics tells us that about two-thirds (actually 68%) of the time, you should expect your actual result to fall within one SD of the EV. Using the numbers above, one SD below EV is -$225 -$806 = -$1031; one SD above EV is -$225 +$806 = +$581.

    Furthermore, about 95% of the time you should expect to finish within two SD's of EV, so in the range of -$1837 to +$1387,
    and 99.7% of the time within 3 SD's of EV, so in the range of -$2643 to +$2193: see https://www.wikipedia.org/wiki/68–95–99.7_rule

    If your results showed a +5% edge over these 5000 rounds at $10/round, then you won $2500: how unlikely is this result for your hypothetical game? To answer this question, we need to calculate the "z-score": how many SD's above EV is the result?

    Well, you expected to finish at -$225, so finishing at +$2500 is $2725 above EV. Since the SD is $806, the z-score is

    Z = +$2725/$806 = +3.38

    Now the Complimentary Cumulative table of the Standard Normal Distribution (see https://www.wikipedia.org/wiki/Stand...ary_cumulative) will tell us the probability of finishing at a given Z or higher. For +3.38, the table gives a probability of 0.00036, or 0.036%.

    Norm's calculator gives 0.03%, so we see that our calculations are pretty close to Norm's. The slight difference is due to two factors: some rounding on my part, and no doubt a slightly different value for the per-round SD (recall I simply assumed a value of 1.14).

    At any rate, we see you have only a tiny chance of being ahead $2500 after flat-betting $10/round for 5000 rounds. How can we improve your chances while sticking to flat-betting? What if you flat bet $500 per round for 100 rounds? That would still be a handle of $50,000, so your EV would still be -$225. However, your SD would be a lot higher:

    New SD = (1.14 units)*($500)*sqrt(100) = $5700

    So your Z would be much smaller:

    New Z = +$2725/$5700 = +0.48

    From the Complimentary Cumulative table, the probability of a Z of +0.48 or higher is about 31.5%.

    Hope this helps!

    Dog Hand

  3. #3


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    1) Is your game Variance = 1.31 ?
    2) what is your bankroll in units ? If you have very small bankroll, you will probably bankrupt FIRST before you reach your goal of 250 units.
    3) I think you should ask : What is the probability to win MORE THAN 250 units after 5000 rounds( flat bet 1 unit each round) ?

  4. #4


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    Hi Dog Hand,

    Are you assuming he have more than 5000 units bankroll ? I think bankroll will affect the probability to reach his goal before he loss his entire bankroll.

    James

  5. #5


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    Great explanation, Doghand!

  6. #6


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    Quote Originally Posted by James989 View Post
    Hi Dog Hand,

    Are you assuming he have more than 5000 units bankroll ? I think bankroll will affect the probability to reach his goal before he loss his entire bankroll.

    James
    James989,

    My calculations are based on an infinite BR.

    If his BR is not infinite, then you are correct. Don S. covers this topic in his seminal Blackjack Attack 3 in his discussion of the "double barrier problem".

    Hope this helps!

    Dog Hand

  7. #7


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    Thanks so much, Dog Hand. I appreciate the time you put into your response. And thank you for showing your work. The answer and interpretation you arrived at were what I thought.

    I used an online sim (I think it was bjsim) a few years back to investigate one of my strategies. (By the way, when I last checked, bjsim had changed some.). Anyway, as I recall it would only let me do sims with small numbers of rounds. When checking one of my strategies. the sim said my E.V. was 5.00% (actually 4.98%) for 5,000 rounds.

    Regarding interpretation: I believe you will all say that the 0.036% probability (please see Dog Hand's post) is meaningless as only 5,000 hands were run in the sim. I no longer have Norm's nice sim that I purchased, so maybe I will ask one of you to run a billion or so rounds if you don't mind.

    But in the meantime, what is the value of all that calculating that Dog Hand did if it is "meaningless." I was under the impression that the z score calculations ALREADY TOOK INTO ACCOUNT the (small) sample size.

    Is it perhaps true that using z scores, etc. for the game of blackjack is an inappropriate use of z scores. etc., unless one is working with, say, 100 million rounds or so or more? But again, though we are only using 5,000 rounds in the above example, standard deviation, etc. is indeed BUILT IN to the calculations, no?, thereby obviating the need for millions of rounds?

    If you believe the 5,000 round probability result (0.036%) is meaningless, does that also mean that you would predict that a sim of one billion rounds with my flat betting, non-progression strategy is no more likely to result in a positive player E.V. than just 'regular' (i.e., no 'strategy' or system) flat-betting, basic strategy Blackjack?

  8. #8


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    The validity of the results from a SIM of 5,000 hands is meaningless. But that has nothing to do with what Dog Hand calculated. His work had nothing to do with a sim, and the results are accurate, no matter what the number of hands.

    Don

  9. #9


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    Thanks, Don, but can you or someone else go deeper please. How exactly does one interpret the result. Please see questions in my most recent post.

    If I am not mistaken, the result of the sim only happens 36 times out of 100,000. In graduate school, we were taught that events more rare than a critical value of 5% suggested that there was something systematic (i.e., something other than chance) at work and that the null hypothesis could be rejected. And 36 out of 100,000 is a whole lot more unlikely than 5 out of 100.

    Again, see address my questions in my post just before Don's post. How EXACTLY does one interpret a finding so uncommon if chance is not responsible, notwithstanding the sample size? If I showed this thread to statisticians, wouldn't they agree something systematic is taking place?

  10. #10


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    There really is no way to answer your question. You say you ran a sim and you gave the results. They are highly unlikely. But never confuse unlikely with impossible. If the sim was correctly run, with no errors, and you got that result, then you got that result. Maybe you did something wrong with the sim. If you didn't, then Dog Hand gave you the likelihood of getting that result. Again, it's extremely small. But that's not the same as impossible.

    Don

  11. #11


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    You must be “lucky”....lol

  12. #12


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    Quote Originally Posted by Overkill View Post
    Thanks, Don, but can you or someone else go deeper please. How exactly does one interpret the result. Please see questions in my most recent post.

    If I am not mistaken, the result of the sim only happens 36 times out of 100,000. In graduate school, we were taught that events more rare than a critical value of 5% suggested that there was something systematic (i.e., something other than chance) at work and that the null hypothesis could be rejected. And 36 out of 100,000 is a whole lot more unlikely than 5 out of 100.

    Again, see address my questions in my post just before Don's post. How EXACTLY does one interpret a finding so uncommon if chance is not responsible, notwithstanding the sample size? If I showed this thread to statisticians, wouldn't they agree something systematic is taking place?
    The sample size doesn't matter. An event having a 0.036% chance of occurring -- it doesn't matter if there is a sample size of 10 or 100,000,000...it's still a 0.036% chance of occurring.

    There are a few ways to go about interpreting this number. One way, you might say the system that is being employed might have some merits, so let's investigate that. Our initial assumption was that the game has a 0.5% player disadvantage (or w/e the number is), but perhaps something we did influenced that. Maybe the player is counting cards or is gaining an advantage in some other way. Maybe the dealer is making mistakes and overpaying the player. In other words, with the initial assumptions this would have a 0.036% chance of occurring, but if the player was <doing advantage play> then this result of +5% profit is actually expected.

    Another way to look at it is that maybe the player just got really lucky. Just because something seems super unlikely doesn't necessarily mean that it is. If you have a 1% chance of something happening, repeat that test hundreds of times and it's very likely to happen. Repeat it many thousands of times, and having it occur back to back is also very likely to eventually happen, even though on any specific instance it's unlikely.

    Something I don't think many look into, but it's also very possible an error was made. If it's a sim on a computer, the program could be written incorrectly. Perhaps you entered the wrong parameters. If you're looking over your gambling diary over a year, perhaps you incorrectly added up the results, over/under estimated your hours or rounds played, or some other type of human error that isn't even directly related to the game itself.

    If it's a sim, you can also do something very easy which is to just run the sim again. And again. And again. The results over all the sims should average out.
    "Everyone wants to be rich, but nobody wants to work for it." -Ryan Howard [The Office]

  13. #13


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    Quote Originally Posted by RS View Post
    The sample size doesn't matter. An event having a 0.036% chance of occurring -- it doesn't matter if there is a sample size of 10 or 100,000,000...it's still a 0.036% chance of occurring.

    There are a few ways to go about interpreting this number. One way, you might say the system that is being employed might have some merits, so let's investigate that. Our initial assumption was that the game has a 0.5% player disadvantage (or w/e the number is), but perhaps something we did influenced that. Maybe the player is counting cards or is gaining an advantage in some other way. Maybe the dealer is making mistakes and overpaying the player. In other words, with the initial assumptions this would have a 0.036% chance of occurring, but if the player was <doing advantage play> then this result of +5% profit is actually expected.

    Another way to look at it is that maybe the player just got really lucky. Just because something seems super unlikely doesn't necessarily mean that it is. If you have a 1% chance of something happening, repeat that test hundreds of times and it's very likely to happen. Repeat it many thousands of times, and having it occur back to back is also very likely to eventually happen, even though on any specific instance it's unlikely.

    Something I don't think many look into, but it's also very possible an error was made. If it's a sim on a computer, the program could be written incorrectly. Perhaps you entered the wrong parameters. If you're looking over your gambling diary over a year, perhaps you incorrectly added up the results, over/under estimated your hours or rounds played, or some other type of human error that isn't even directly related to the game itself.

    If it's a sim, you can also do something very easy which is to just run the sim again. And again. And again. The results over all the sims should average out.
    RS, that +.5% or1% sort of disappears when you don’t know all the deviations, don’t always bet optimally, sometimes get distracted, play fatigued etc, does it not?

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