Variance in Blackjack

Sorry for the generic name but here's the situation:

At one time in my life I knew this stuff. I have since lost that knowledge and think I've discovered it but am not too sure. I cobbled together information from the Wizard's multi-line video poker page and blackjack variance page. Additionally, I used:

http://www.jazbo.com/videopoker/nplay.html

As in the example... one standard deviation for 1200 hands of 10/7 is sqrt(1200*28.257)=184 bets. Is this directly transferable to blackjack?

500 hands
variance = 1.15 (let's just assume it's that)
EDIT: For anyone reading this for the first time, 1.15 is the standard deviation. Variance of BJ is approximately 1.15^2 or 1.32.

sqrt(500*1.15)=24 bets
three standard deviations... 89%... 72 bets
EDIT: Standard deviation, I'm using a smaller confidence interval : (1-1/(# std deviations)^2) so 3 standard deviations is 1-1/9=8/9 instead of 99.7%.

So if I'm flat betting quarters... after 500 hands... I'm going to win or lose more than $1800 off of the expected value only 11% of the time?

Quote: ahiromu

...As in the example... one standard deviation for 1200 hands of 10/7 is sqrt(1200*28.257)=184 bets. Is this directly transferable to blackjack?

500 hands
variance = 1.15 (let's just assume it's that)

sqrt(500*1.15)=24 bets
three standard deviations... 89%... 72 bets

So if I'm flat betting quarters... after 500 hands... I'm going to win or lose more than $1800 off of the expected value only 11% of the time?

The probability that you are worse off more than 3 standard deviations below the mean is not at bad as 11%--it is only 0.13%. (By the way, for 2 standard deviations it's 2.28%, and for 1 standard deviation it's 15.87%.)

Also, for blackjack the variance is about 1.32, so the standard deviation is about 1.15. So the standard deviation for 500 hands would be sqrt(500*1.32) = 25.7 bets. Then 3 standard deviations for 500 hands would be 77 bets.

Using 0 as the mean is a good approximation, but for a basic strategy player at an S17 game the mean for 500 hands would be about 500*(-0.5%)= -2.5 bets.

Regarding the standard deviations, would you have a comment on the linked article's explanation of standard vs. non-standard result distributions? I understand how you get your numbers, but this was another thing that I was having an issue with.

Thanks for correcting me with regards to variance vs. standard deviation.

Yeah, I've decided that using 0 as the mean is good enough. I'm not calculating thousands of hands.

Quote: ahiromu

Regarding the standard deviations, would you have a comment on the linked article's explanation of standard vs. non-standard result distributions? I understand how you get your numbers, but this was another thing that I was having an issue with...

Now I see where your 11% figure for 3 standard deviations from the mean came from. I admit that I am a person who, in the words of the article, "uses the normal distribution as a yardstick to make statements about expected results relating to standard deviation." I will have to defer to the mathematicians on this board on the correct way to estimate the confidence intervals for video poker results.

Alright, so the answer is sorta 11% and sorta .13%? The fun math. Maybe someone will be able to clear up this figure for us.

Splitting your bet into two bets doesn't help as much as I thought it would, that's unfortunate.

Quote: link

By the way, I see many people using the normal distribution as a yard stick to make statements about expected results relating to standard deviation. Remember that for the number of hands you have in a session, the distribution of results is decidedly non-normal. Instead of estimating the result as 95% confidence of being plus or minus two standard deviations (which would be true for a normal distribution), an estimate of being within (1- 1/(# std deviation)^2) is more appropriate. This says you should expect to be within +- two standard deviations 75% of the time, and within +- 3 standard deviations 88.9% (not 99.7%).

I have some questions about this too.

A#1, why is a normal distribution not to be expected and when is it not to be expected?

Quote:

if you play the same game in a 4-play version, then the variance will be higher

Is this what he is talking about?

To further complicate it, the 11% becomes 5.5% for the chances of outlier variance on the 'bad' side.

PS:

In answering one question, I see the Wizard was happy with using Normal Distribution except for his video poker example.

[the second question]

https://wizardofodds.com/ask-the-wizard/287/

I think I'm happy with using normal distribution, I understand statistics and as long as I don't set a ridiculous goal (such as having a 5% chance at losing my entire bankroll) I'll be fine. Splitting eights four times, doubling on each, then losing, CAN happen, so you want to prepare for that with these smaller confidence intervals. At least that's what I'm getting out of it.

Wanted to follow-up.

sqrt(1200*1.32) = 40 units
3 standard deviations is 120 units

Flat betting 1 unit
Perfect basic strategy

In personal experiences, will I be able to play 1000 hands with 120 units? I'm asking because the above calculation does not take into account the possibility of busting your bankroll. If I wanted to quit at losing 120 units, do you guys think that I could last a thousand hands?

$5 units, $600 bankroll
$10 units, $1200
$25 units, $3000