optimal betting fraction for non-bj game

Suppose a game that has 0.0001 chance of winning 11000 units and

0.9999 chance of losing 1 unit

thus 0.0001*11000 - 0.9999*1 yields +0.1001
How many units should I bet if I have 500,000 units bank.

Second game that has 0.0003 chance of winning 11000 units and

0.9997 chance of losing 1 unit

thus 0.0003*11000 - 0.9997*1 yields +2.3003
How many units should I bet if I have 500,000 units bank.

Third game that has 0.0001 chance of winning 11000 units and

0.9999 chance of winnng/losing X unit,

where X has a mean of -1 and SD of 150

(ie: X is between losing 151 and winning 149 two thirds of the time.)

thus 0.0001*11000 - 0.9999*1 yield +0.1001
How much units should I bet if I have 500,000 units bank.

If anyone knew the formula to calculate the above queries, please paste them. Otherwise can anyone give me links to references/literatures on non-even-money payoff (advantage) game please.

Your queries can be reduced to the basic question-what is the correct way to bet Kelly with non-even payoffs? You probably already know that the Kelly criterion recommends betting the same % of your bankroll as your % advantage on a given wager, and is the best compromise between risk and return.

For non-even payoffs you just divide your Kelly wager between the ratio of a winning wager to a losing wager. So, a 10% edge with a 10/1 payoff would make your kelly bet 1% of bank, .1% with a 100/payoff, and (borrowing an interesting example from the bjmath pages), 100% with a 1/10 payoff.

In the first example, its obvious your return on investment is so small its not worth bothering with due to practicality. In the second, if I have understood your meaning correctly I get a 59 unit bet, which is again so small its not really worth bothering with given the return on investment. You also need to know that few professional card players bet as much as full Kelly due to a range of practical factors which may or may not apply to your examples.

In the third example I believe you have made an error when you typed "0.9999 chance of winnng/losing X unit"-please clarify. In any event, calculating SD is not neccessary for computing kelly bet size.

Generally speaking, with the extreme longshots you are talking about it is not worthwhile betting due to the conservative betting neccessitated by the high probability of a loss on each trial, unless some means of collapsing the envelope of possibilities exists.

The best reference on this subject is Thorp's "The Mathematics Of Gambling", which is free at the aforementioned bjmath site.