Variance for multiple hands

Scenario a : 1 hand x $B, variance = V1, expectation= EV1

Scenario b : n hand x $B/n, combine variance for n hands = V2, combine expectation for n hands = EV2.

EV1 = EV2, how about the relationship between V1 and V2 ?

If the results of the hands in scenario b are independent of each other:

* V2_one mini hand = V1/(n^2)
* Total variance of scenario b, V2 = n*V2_one mini hand = n*V1/(n^2) = V1/n

scenario a: V1
scenario b: V1/n

However in most casino game the results of the hands at a single table in a single round are significantly correlated. This means the problem is way more complex and varies by game.

Are you sure V2 = n*V2_one mini hand ?

Quote: ssho88

Are you sure V2 = n*V2_one mini hand ?

Yes, given the hands are independent. http://en.wikipedia.org/wiki/Variance#Basic_properties

But they aren't independent in blackjack, so unfortunately it's not that easy.

The key formula here is:

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

where:

Cov(X, Y) = E(XY) - E(X)E(Y)

and

Var(X) = Cov(X,X) = E(X^2) - (E(X))^2.

If Cov(X,Y) = 0 then the variables are said to be uncorrelated, and Var(X + Y) = Var(X) + Var(Y). Independent variables are uncorrelated.

Quote: ssho88

Scenario a : 1 hand x $B, variance = V1, expectation= EV1

Scenario b : n hand x $B/n, combine variance for n hands = V2, combine expectation for n hands = EV2.

EV1 = EV2, how about the relationship between V1 and V2 ?

i need this kind of data, so i tried to test it by myself.

the relationship is N is bigger, the V2 is smaller. in other words, the more hands(tables, investments...) you play, the covariance is smaller. or playing more hands or do more different investments+team-money-sharing will reduce the variance/increase the kelly/multiply the profit.

let's say:
1%EV
2hands 3hands 4hands 5hands 6hands 7hands
1.4 1.8 2.1 2.3 2.4 2.8 (here it is strange, i used 500,000 random numbers, each 100 hands/set, 10 times, then choose average).

here note if EV is different, the result is different.

1.2%EV
2hands 3hands 4hands 5hands 6hands 7hands
1.4 1.8 2.0 2.2 2.4 2.5

5%EV
2hands 3hands 4hands 5hands 6hands 7hands
1.4 1.6 1.8 1.9 1.9 1.9

10%EV
2hands 3hands 4hands 5hands 6hands 7hands
1.2 1.2 1.0 1.3 1.1 1.1

when 10% EV, the covariance is even smaller than 2 hands.

by this kind of the result, we can say we need to cooperate with others no matter how rich or capable we are.