Optimal betting of multiple hands

[Eric]

I've been trying to figure out the optimal bet for blackjack, given a certain number of hands and my advantage. I think my math is wrong, because I'm getting some funky results. Does anyone know the correct math for this? Read on for my attempt.

I'm using the kelly function:
bet = advantage * bankroll / variance

Assume my bankroll is 1000, my advantage is 0.05, and the variance of 1 hand is 1.32.

bet = 0.05 * 1000 / 1.32 = 38

Now, you would assume that it would be be better to spread those 38 bets to 5 hands, rather than have them all on the same hand.

Computing the bet again, for 5 hands this time, I get some weird results. The variance of 5 hands of blackjack is 16.20. (using the function 1.32*n + 0.48*n*(n-1) )

bet = 0.05 * 1000 / 16.20 = 3.09

3.09*5 hands = 15. Shouldn't this number be greater than 38?

Should I be dividing by standard deviation instead of variance?

 

[KenSmith]

Explained

The variance you calculated for all 5 hands is fine, but you then need to divide it by 5 before using it in your bet-sizing formula, which expects the variance per bet.

So, the average variance per bet for each of your five bets is 16.20 / 5 = 3.24.

Your optimal bet on each hand is 0.05 * 1000 / 3.24 = 15.43.

Your overall action of 15.43 * 5 = 77.15, which is as expected, quite a bit larger than the 38 units you should bet on a single hand.

P.S. For anyone following along, the magic numbers in the equation are variance=1.32, covariance=0.48.