Suppose I'm playing 2 hands and want to have $200 total out at a specific count. What is the difference in ROR if I split the hands $100 each vs. putting $50 on one and $150 on the other?
TIA
The latter has greater ROR because the overall variance is greater. If you have a certain sum to bet over two hands, the total variance for the two hands will always be the smallest if the two sums are equal. As soon as they are unequal, the variance increases. And since the total dollar amount is the same, which means the e.v. is the same, the ROR is greater if the variance is greater.
As to how much greater the variance and ROR become for unequal hands, the formula is a bit complicated, but I have it if you really need it. Suffice it to say that, the more there is on a single hand, the greater the variance. Clearly, if you have $199 on one hand a $1 on the other, that's going to create the biggest variance for the $200 total.
Don
Suppose variance of a single bet is var; covariance between two simultaneous bets is cov; bet size 1 is b1; and bet size 2 is b2. Then the total variance of the two simultaneous wagers is:
var(b1^2 + b2^2) + cov[(b1 + b2)^2 - (b1^2 + b2^2)]. In words: Total variance is: variance times the sum of the squared bets, plus the covariance times (the entire expression of) the square of the sum of the bets minus the sum of the squares of the bets. (And, of course, the overall s.d. for the two hands is the square root of the total variance.)
Homework: For var = 1.33, cov = 0.50, b1 = $150 and b2 = $50, calculate the total variance (and then the s.d.) of the two simultaneous hands. Then do it again if the two bets are even (i.e., $100 each). Finally, we note the trivial: the s.d. for a single hand of $200 is 1.15 x $200 = $230; so the two s.d.s that you find, above, need to be smaller than that, and the s.d. for the two equal hands of $100 each needs to be smaller than the s.d. of the two unequal hands of $50 and $150.
Clear?
Don
Logistical problems with this morning's homework assignment. No, the dog didn't eat my homework, nor did I leave my book in a hotel room. I couldn't find my calculator in my desk drawer. Solved that using the Windows calculator. Then, tried to write it out on a small piece of scrap paper and ran out of room
In any case. when betting 150/50, var = 40750 and SD = 201.87. When betting 100/100, var =36600 and SD = 191.31.
Thanks for your help. This was for an AP friend of mine who is not on the forum.
Last edited by 21forme; 02-01-2020 at 06:48 AM. Reason: caught a typo before Don saw it
Because that's the covariance for two simultaneous hands of blackjack! (What else could the answer possibly be?)
Might it help if you specified what the other game was? Or, do you want it for every other game under the sun? In any event, for me, at least, it's academic, because I don't have the slightest idea for other games.
Don
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