What is the variance of multi-hand bet vs single hand flat bet?

In blackjack.

For example. What is the varience on a $1000 bankroll of:

1: Single hand $5 bet at a time
2: 3x hands at $5 each
3: Single hand $15 bet at a time

Obviously 3x $5 should have variance somewhere between 1x $5 and 1x $15. But where exactly it fits is beyond me at the moment.

Does anyone know?

Quote: Neutrino

For example. What is the varience on a $1000 bankroll of:

1: Single hand $5 bet at a time
2: 3x hands at $5 each
3: Single hand $15 bet at a time

Obviously 3x $5 should have variance somewhere between 1x $5 and 1x $15. But where exactly it fits is beyond me at the moment.

Does anyone know?

It depends on the covariance between the hands, which depends on the game.

I forgot to specify it was blackjack. I'm minorly interested how other game have their varience affected by multi-hand but i'm mostly interested in blackjack.

Quote: Neutrino

I forgot to specify it was blackjack. I'm minorly interested how other game have their varience affected by multi-hand but i'm mostly interested in blackjack.

I don't know the answer off the top of my head. I'm sure it varies as the rules change.

The general formula is:

Var(X + Y + Z) = Var(X) + Var(Y) + Var(Z) + 2 Cov(X, Y) + 2 Cov(Y, Z) + 2 Cov(X, Z). Since the bets are all the same, this reduces to:

3x + 6y, where x is the variance of one hand and y is the covariance between any two different hands.

Also note that variance is a squared value, so in particular Var(3X) = 9 * Var(X), so this gives a result that is less than 1 hand of triple the bet. (In this case, Var(1 hand of $5) = x, Var(1 hand of $15) = 9x, and Var(3 hands of 5) = 3x + 6y. 0 < y < x, so you are correct in your intuition that it lies between the two values.

As it is quite clear that var($5 hand) is lower than var($15 hand), I think the interesting question should take into account the OP's mention of a $1000 bankroll.

But for this, we must know how he plays that bankroll. Too simple to imagine he plays 200 $5 hands. There are splits and doubles. He probably plays some of the won money, too.