SD when playing multiple hands

How do you calculate the SD on multi handed blackjack if you know the SD on one hand? I.e. if you re playing $30 a hand and the SD per hand is .44, what would be the SD per round playing two hands of $30? What about three hands of $30 per round?

Quote: Swanson234

How do you calculate the SD on multi handed blackjack if you know the SD on one hand? I.e. if you re playing $30 a hand and the SD per hand is .44, what would be the SD per round playing two hands of $30? What about three hands of $30 per round?

Not enough information; you also need to know the covariance between the hands.

Variance is the square of standard deviation, and

Var(X + Y) = Var(X) + Var(Y) + 2Cov(X, Y).

So, square the standard deviation, double it, add double the covariance, and take the square root to get the standard deviation of two hands.

there are tables out there that have a sample cov calculated from a simulation for different bj rules . the cov is positive so suffice to say the SD is higher for multi-hand as compared to playing hands back to back..

when you find the cov you can calculate the total variance n*(var+(n-1)*cov).

Isn't the SD of 2 hands of $30 equal to the SD of betting one hand of $43 (72% of $60) per round?


SD of three bets of $30 equal to SD of one bet of $60 (2/3 of $90) per round?

Something like that.

Quote: Swanson234

Isn't the SD of 2 hands of $30 equal to the SD of betting one hand of $43 (72% of $60) per round?


SD of three bets of $30 equal to SD of one bet of $60 (2/3 of $90) per round?

Something like that.

the SD of n independent hands is sqrt(n)*(SD)... so your first conclusion is correct (if you are playing the hands against different sets of dealer cards) and your second is not.

note: multiple multi-hand deals are independent. so you can multiply the multi-hand column by sqrt(n)*SD to figure out the total SD for multiple multi-hands

n (~SD multi-hand) (~SD played against separate dealer hands)
1 35 35
2 57 49
3 79 60
4 101 69
5 122 77
6 144 85
7 165 92
8 186 98