Betting Progressions

If you need to win 1 hand (without losing the hand) to increase your bankroll to a desired amount and have only 1 hand to do it in then the probability of success is .44 (Using Wong's figures for W = .44 , P = .08 , L = .48)

However if there are several hands to try for the win then, because of the push possibility, the probability of success can rise to a high of .4783

For example there are 4 hands left to play then the number of ways to achieve the goal (without losing a hand) is as follows:

W............................... = .44.......................................... = .4400
P W............................ = .08 x .44................................... = .0352
P P W......................... = .08 x .08 x .44............................ = .0028
P P P W....................... = .08 x .08 x .08 x .44.................... = .0002

.....................................................................Total........ = .4782



Using Wong's progression of 1/15 , 1/7 , 1/3 , 1 (The Martingale betting system for doubling up the bet after each loss) and there are 4 hands to accomplish the goal and you can stand 2 losses then the probability of success is .8530 as shown below.

W............................... = .44........................................... = .4400
P W............................ = .08 x .44.................................... = .0352
L W............................ = .48 x .44.................................... = .2112
P P W......................... = .08 x .08 x .44............................. = .0028
P L W......................... = .08 x .48 x .44............................. = .0169
L P W......................... = .48 x .08 x .44............................. = .0169
L L W......................... = .48 x .48 x .44............................. = .1014
P P P W...................... = .08 x .08 x .08 x .44...................... = .0002
P P L W...................... = .08 x .08 x .48 x .44...................... = .0014
P L P W...................... = .08 x .48 x .08 x .44...................... = .0014
P L L W...................... = .08 x .48 x .48 x .44...................... = .0081
L P P W...................... = .48 x .08 x .08 x .44...................... = .0014
L P L W...................... = .48 x .08 x .48 x .44...................... = .0081
L L P W...................... = .48 x .48 x .08 x .44...................... = .0081

..................................................................Total............ = .8530

(The column as shown adds up to .8531 but more of the entries were rounded up than down)


Below is calculated the various probabilities of success for hands left to play vs losses allowed. I hope players can put these to use.


Hands...................<----------------losses tolerated------------------>
..............................0...............1...............2...............3.............4...
...1..................... .4400........ .4400........ .4400........ .4400........ .4400
...2..................... .4752........ .6864........ .6864........ .6864........ .6864
...3..................... .4780........ .7230........ .8244........ .8244........ .8244
...4..................... .4782........ .7273........ .8530........ .9017........ .9017
...5..................... .4783........ .7277........ .8573........ .9216........ .9449
...6..................... .4783........ .7278........ .8579........ .9252........ .9579
...7..................... .4783........ .7278........ .8580........ .9258........ .9608
...8..................... .4783........ .7278........ .8580........ .9259........ .9612

 

Thanks

This stuff is good to have worked out - thanks Bluelight -

 

This is an area of strategy I have done extensive research into. Kudos to you for analyzing it in some depth. While my own findings will not be public until my tournament book is completed, I would encourage you to press farther and realize that in tournaments, it is so often less about win/win/lose/etc and more about net chip gain. Therefore, one must assign a probability of achieving blackjacks and key splits/doubles when in the later stages of a progression, and incorporate this additional net chip value into the total probability of success for each progression (adjusting, of course, for a lower tolerance towards sub-optimal doubles/splits on higher steps of the progression). Then the true power and value of this type of tournament move can be completely quantified.

-hd.