I have quite a bit more information than this if needed, but my basic question is about the probability of a lucky run like that.

Thank you for your time.

Quote:FollowCan anyone help me understand the math for a player to turn a +1.3% EV in blackjack over the course of nearly 20,000 hands without cheating? Player bets an average of $163 per hand and is winning around $2.14 per hand over 20,000 hands.

I have quite a bit more information than this if needed, but my basic question is about the probability of a lucky run like that.

Thank you for your time.

link to original post

Your question is not worded well enough for us to answer cogently.

1. Are the basic rules of the game so tilted towards the player that he has a +1.3% EV? I don’t think that’s what you meant.

2. The RESULTS which you describe are NOT the EV, rather, they are the RESULTS. The EV would have been determined by knowing the exact rules of the game, which you haven’t supplied.

It seems like you are saying ‘you’ won around $43k betting $3.26 million. If my math is right, your results are pretty darn good, but certainly (most likely) not extraordinary.

Hopefully I’ve been helpful. Welcome to the forum.

Game rules:

Dealer hit soft 17

Double any

Split once

Blackjack plays 3:2

Continuous shuffle

No surrender

I took the standard ROI of 99.5 for every 100 bet as my base and found the return on this run of 20k hands to look fairly extraordinary to me. Can you use this information to tell me the mathematical likelihood of this kind of run? If it matters, this is over the course of 25 sessions beating the negative almost every time, even in the few losing sessions. I have each session entered on a spread sheet if that will help you.

There is a formula for working out the variance and standard deviation for a combination of wagers. But it goes to hell in a handcart unless the wagers are all the same. You can't just plug in the average wager.Quote:FollowCan anyone help me understand the math for a player to turn a +1.3% EV in blackjack over the course of nearly 20,000 hands without cheating? Player bets an average of $163 per hand and is winning around $2.14 per hand over 20,000 hands.

I have quite a bit more information than this if needed, but my basic question is about the probability of a lucky run like that.

Thank you for your time.

link to original post

If you increased bankroll X into Bankroll Y using a LOW house edge game and ANY betting pattern, you can estimate how likely that was with my rule of thumb. P<=X/Y

E.g if you had the objective of turning $1000 into $1010 using $1 base martingale, with the only acceptable outcomes being success or loss of all bankroll, then P<=1000/1010 = 99%

https://wizardofvegas.com/member/oncedear/blog/7/#post1370

So if you meant that you made a 1.3% profit having put your bankroll at real risk, then P<=100/101.3 or about 98.7% likely

Quote:FollowCan anyone help me understand the math for a player to turn a +1.3% EV in blackjack over the course of nearly 20,000 hands without cheating? Player bets an average of $163 per hand and is winning around $2.14 per hand over 20,000 hands.

I have quite a bit more information than this if needed, but my basic question is about the probability of a lucky run like that.

Thank you for your time.

link to original post

You are very lucky, I would guess that you are about 2.5 Standard Deviations above the average, if you were flat betting (obviously you weren't).

Quote:OnceDearThere is a formula for working out the variance and standard deviation for a combination of wagers. But it goes to hell in a handcart unless the wagers are all the same. You can't just plug in the average wager.Quote:Follow

I have quite a bit more information than this if needed, but my basic question is about the probability of a lucky run like that.

Thank you for your time.

link to original post

If you increased bankroll X into Bankroll Y using a LOW house edge game and ANY betting pattern, you can estimate how likely that was with my rule of thumb. P<=X/Y

E.g if you had the objective of turning $1000 into $1010 using $1 base martingale, with the only acceptable outcomes being success or loss of all bankroll, then P<=1000/1010 = 99%

/member/oncedear/blog/7/#post1370

So if you meant that you made a 1.3% profit having put your bankroll at real risk, then P<=100/101.3 or about 98.7% likely

As stated, this is over the course of 20,000 hands and 25 sessions. The bets are almost always a standard 150 with the other 13 of the average coming from double downs, splits, etc...

Quote:ksdjdjYou are very lucky, I would guess that you are about 20 Standard Deviations above the average, if you were flat betting (obviously you weren't).

The bets were flat and I realize that the run is absolutely improbable. I want to know how improbable such a run is mathematically. I'm good with numbers and statistics, but figuring out this particular math is beyond my skill, which is why I'm turning to this forum and hopefully those better versed with math than I am.

Quote:Follow

I have quite a bit more information than this if needed, but my basic question is about the probability of a lucky run like that.

Thank you for your time.

link to original post

For 20,000 hands, your profit is about (20,000)(2.14) = $42,800.

For your rules, the Wizard of Odds gives a house edge of about 0.7%, so let's use that instead of 0.5%. Your expected loss on 20,000 hands of $150 initial bet size would be (0.007)(20,000)(150) = $21,000.

I couldn't find the exact variance for your rules, but suppose for a 1-unit-bet hand of blackjack it is 1.346. Then the variance for playing 20,000 hands at $150 per hand would be (150)

^{2}(20,000)(1.346) = 605,700,000.

The standard deviation would be the square root of the variance and would equal about $24,611.

The number of standard deviations that your result is above the expected loss is: ($42,800 + $21,000) / $24,611, which is about 2.59.

The probability of being at least 2.59 standard deviations above the mean in a normal distribution is about 0.48%.

So, it looks like your luck is rare but not impossible.

Quote:FollowQuote:ksdjdjYou are very lucky, I would guess that you are about 20 Standard Deviations above the average, if you were flat betting (obviously you weren't).

The bets were flat and I realize that the run is absolutely improbable. I want to know how improbable such a run is mathematically. I'm good with numbers and statistics, but figuring out this particular math is beyond my skill, which is why I'm turning to this forum and hopefully those better versed with math than I am.

link to original post

Short answer, you had about a 1% chance of this happening (rough guess).

Note: I don't know why it says 20 SD in my original answer because I cut and pasted it as 2.5 SD (20 is obviously wrong, sorry).

Last year, 2020, my expectation for my blackjack play was more than $80,000. Since I play +EV (card counting), this $80,000 represents what I call accumulated EV of each session. So over $80,000 in expectation. My actual results a little better than $14,000. That would be $65,000 below expectation. In terms of rounds played that was about 40,000 rounds, so double what you are talking about.

And it goes both ways. This year I was on the complete opposite end, some $50,000 above expectation for the year.

Now these two years for me have been the most extreme I have ever had and weirdly they occurred back to back, one in each direction.

Your results are in the same ballpark. Since your expectation was negative (not sure how much) it looks like you finished $30,000, or $40,000 above expectation for those 20,000 rounds. Unusual yes. But the unusual happens with variance sometimes. And really nothing is unusual until you get well above 3 standard deviations. That is when you are really starting to talk unusual results.

So I wouldn't think any more about it that you finished with good positive variance. Enjoy it. :)