Multiple Hands Variance in $$

A

assume_R

Well-Known Member
Can somebody please check my math and let me know where I am mistaken? And I'm going to assume that the frequencies of each count don't matter, "eating up cards" don't matter, etc. etc.

Situation: EV = +1%, Var = 1.3, Cov = 0.5, Bet = $20

Situation #1: 1 hand of $20
EV = 1% * $20 = $0.20 / round.
Var = 1.3 * $20^2 = $^2 520 / round. Std = $23 / round

Situation #2: 2 hands of $10 each
EV = 1% * $10 + 1% * $10 = $0.20 / round
TotalVar = Var * n + Cov * n * (n - 1)
TotalVar = 1.3 * 2 + 0.5 * 2 * 1 = 3.6
TotalVar = 3.6 * $20^2 = $^2 1440 / round. Std. = $38 / round

Shouldn't the std be less when 2 hands are played with the same $$ on the table?? Perhaps my last line should have read 3.6 * $10^2??? which would make the std. $19 / round??
 
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assume_R said:

Situation #2: 2 hands of $10 each
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TotalVar = 1.3 * 2 + 0.5 * 2 * 1 = 3.6
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Click to expand...

Don't multiply the variance by 2. The total variance for each hand should be 1.3 + 0.5 = 1.8.

-Sonny-
 
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Thanks Sonny.

Yet that means that the EV is the same both situation #1 and #2 ($0.20), yet the variance for situation #2 (multiple hands) is 1.8 * $20^2 while the variance of situation #1 is 1.3 * $20^2? Or do I multiply that 1.8 by $10^2?

I only did what I did because of:

"According to Professional Blackjack by Stanford Wong (page 203), the variance for similar rules is 1.32 and the covariance is 0.48. The total variance of n hands would be 1.32*n + 0.48*n*(n-1). Take the final square root to get the standard deviation."

from http://wizardofodds.com/blackjack/appendix4.html

So what about if there are 3 hands??? The standard deviation according to wizardofodds would be sqrt(1.3 * 3 + 0.5 * 3 * 2) = 2.63

Sorry for the confusing post I'm just trying to make sure I understand this 100%.
 
Okay I think I finally answered my own question from http://www.bjmath.com/bjmath/ror/tripror.htm (Archive copy)

1. Variance = 1.3 * n + 0.5 * n * (n-1)
2. Variance/hand = Variance / n = 1.3 + 0.5 * (n - 1)
3. Variance/hand in $$^2 = [1.3 + 0.5 * (n - 1)] * (Bet/hand)^2
4. Variance/round = Variance/hand * n = Var/hand * n = [1.3 * n + 0.5 * n * (n - 1)] * (Bet/hand)^2
5. Std/round = (Bet/hand) * sqrt(1.3 * n + 0.5 * n * (n - 1)) for n hands

So in my case, std/ round = $10 * sqrt(3.6) = $19 / round while my EV is $0.20 / round

Regarding Sonny's post, I see now that 1.8 is per hand, with a $10 bet. I multiplied by 2 to get the variance per round. But that $10, though, should stay at $10 (not $20) based on the equations i derived above.
 
The Important Points

If you bet approx 73% over each of 2 hands that you would bet on 1 hand then your long term ror is the same.

Example:
Instead of one hand of $100 you can bet two hands of $73 or $75 and have the same approx long term ror.

However:joker::whip:

If playing 2 hands over a session you will need more cash in order to keep the trip ror the same, approx 20% to 33% more $ then you would need if you were to just play one hand.:joker::whip:
 
assume_R said:
Regarding Sonny's post, I see now that 1.8 is per hand, with a $10 bet.
Click to expand...

Right. So for 3 hands the variance would be (1.3 + 0.5 * 2) = 2.3 per hand. For all 3 hands the SD would be sqrt(2.3 * 3) = 2.63, which confirms your numbers above. My version just rearranges the formula a bit:

(1.3 * 3) + (0.5 * 3 * 2) = (1.3 + 0.5 * 2) * 3

-Sonny-
 
Sonny said:
Right. So for 3 hands the variance would be (1.3 + 0.5 * 2) = 2.3 per hand. For all 3 hands the SD would be sqrt(2.3 * 3) = 2.63, which confirms your numbers above. My version just rearranges the formula a bit:

(1.3 * 3) + (0.5 * 3 * 2) = (1.3 + 0.5 * 2) * 3

-Sonny-
Click to expand...

Okay, thanks for confirming for me. I finally got it down pat. I was confused in part also because on your previous posts (from 2009 I think) I never noticed you including the (n-1) term. I think we're on the same page now, and the equations are just rearrangements of each other.

And P.S. thanks for gettin rid of that spammer haha
 
Perfectly understandable. However, the equation in the post you linked to (BR * EV / Var = Bet) doesn't take into account the fact that you might also play some -EV hands.

To take that into account, you must multiple that number ("Bet" above) by a factor which Brett Harris calls "kb". And to calculate kb, you need to know your total variance at max bets. So if your max bet is at TC = +5, and you always play 2 hands at TC = +5, you'd need to know that total variance for the TC = +5 round, which is why I wanted it in that form instead of per hand.
 
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