#1




Optimal betting of multiple hands
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*(n1) ) 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? 
#2




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 betsizing 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. 
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