Fairly Straight Gambling Math Question

[flatbush]

Howdy, Fellow players. I have a fairly straightforward math question that can be applied to any game. If a player has a + edge of: let's say a true 1/2 % in a BJ game but only during certain circumstances and those circumstances occur only on average once every 4 hands. If the player were to raise his bet for those special situation hands to let's say 10 times his normal flat bet unit, that he makes during the other 75% of the time, what would his "effective edge" be over the course of many hands played?? The way I figure it is, and I could certainly be wrong ( why I am asking for help) is like this: In 100 hands 75 will be played at 1$ and all those bets carry a negative edge against the player of minus 1/2%, so 75 X $1.00 X - .005 = - 37.5 cents over 75 hands versus the gain which is expressed as 25 X $10 X +.005 = + $1.25 so overall in the average batch of 100 hands the result at these $ levels would be $1.25 - .37.5 = + 87.5 cents.

Here then are the 2 questions that arise from this assuming that my math is correct: What is the positive edge in % terms that such a player enjoys? Secondly how would one calculate their bankroll for risk of ruin purposes? Would the bankroll be sized for the inherent edge of + 1/2 % or would you size the B/R for the overall percentage edge generated by the whole scheme ?? Thanks a million, Flatbush

[Dog Hand]

flatbush,

The "positive edge in %", which is called the "Initial Bet Advantage" or IBA, is the average gain divided by the average initial bet. In your hypothetical example, the average gain per round is $0.875/100 = $0.00875, and the average initial bet is (75*$1 + 25*$10)/100 = $3.25, so the IBA is $0.00875/$3.25 = 0.00269... or almost +0.27%. Finally, this is the IBA (also called the "Expected Value", or EV) that you would use in your Bank Roll calculations.

Hope this helps!

Dog Hand

[Gronbog]

In order to calculate risk of ruin, you also need the standard deviation of both situations.

[flatbush]

Dog Hand Thank you so much for the thorough breakdown of the EV. or IBA. The final question I would have is when one calculates their needed bankroll (in order to have an X % chance of being able to play perpetually) given a certain positive edge, the answer comes back in Y units needed to accomplish that feat. The final question is when you consider units do you consider it is the Large 10 X wager or the average wager that you make ?? Thanks so much again, Flatbush

[DSchles]

The final question is when you consider units do you consider it is the Large 10 X wager or the average wager that you make ??

There was a third choice! So, the answer is neither. The unit is the small bet.

Don

[DSchles]

In order to calculate risk of ruin, you also need the standard deviation of both situations.

If it's blackjack, let's just make it 1.15 for both types of wager.

So, for the example under discussion, s.d/hand is sqrt[(1^2*0.75*1.15^2) + (10^2*0.25*1.15^2) = 5.84. And the variance is the square of that, or 34.05

And e.v./hand is the already determined 0.00875 from above.

Next step is to determine the required bankroll for whatever ROR makes you happy. Suppose you choose 5%. You can use Norm's calculators, above, or the formula on p. 113 of BJA3, to wit: B = -(var/2e.v.)ln ROR. In this case: B = -(34.05/0.0175)*(-3.0) = 5,829 units.

Don

[flatbush]

Don, Thank you so much for this rather eye opening analysis of my question, and it's a little sobering to see how many units would be required for a good chance of perpetual play. I will print out this math and keep it for future reference. Thanks again, Flatbush