Hi,
Bankroll in Units = 400
Win rate per 100 hands =1
Std. Dev. per 100 hands =22.58
Hands Played = 20000
The ROR calculated by the RISK CALCULATOR = 8.543%
Anyone/Don/Norm can show me how to manually calculate the ROR ?
James
Don,
On page 133. the ROR formula consists of two terms, the above example is referred.
First term, P(X<-400) = P(Z< (-400-0.01*20000)/2.258/(20000)^0.5) = P(Z< -1.879), Prob = 3.01%
P(Z< (-400+0.01*20000)/2.258/(20000)^0.5) = P(Z<-0.262), Prob = 0.2657
Second term = e^(-2*0.01*400/2.258^2)*0.2657 = 0.05533 = 5.533%
So total ROR = 3.01 + 5.533 = 8.543%, Correct ?
Total ROR = ROR after reach the trip length( first term) + ROR before the trip length(second term) , is this statement correct ?
First term is understandable, it is the ROR after you reached 20000 hands and I agree with the NORMAL DISTRIBUTION FUNCTION.
As for second term, you used a fraction of LONG TERM ROR to cover the ROR before you reach 20000 hands, and I guess the fraction(0.2657 in this example) is for game with negative ev = -0.01( favour to the dealer), may I know what is the logic behind ? I am not questioning the accuracy of the formula but just want to know the fundamental of that formula.
Hope can learn something new from you. Thanks
Did you read the bottom of 132 and to p of 133? I can't tell you how the fraction of the long-term ROR portion was derived, but I don't see why you assume a game with negative e.v. There's only one game you're playing.
The formula was derived from the one used for (one-touch) barrier options, but I am not able to tell you how to derive it. Sorry.
Don
Thanks for your reply.
I found that the long term ROR will be much lower than the expected 13.53%(one kelly bankroll and betting) for game with relatively high ev(>10%) and high S.D(>4), this was confirmed by simulations and LONG TERM RISK CALCULATOR.
Here is a real game with ev = 9.46%, S.D = 3.685, BANKROLL = 144 units(one kelly bankroll),
1) ROR(long term risk calculator) = 12.85%
2) ROR(risk calculator with time constraint, 1000000 rounds) = 13.45%
3) ROR(my own simulations) = 12.01%
Question 1 : I thought long term ROR should be 13.53%(compare to 12.85%) ?
Question 2 : I thought trip ROR should be LOWER than long term ROR ? Why trip ROR is higher ?
Any comments ?
1. For (1), it would appear that the calculator uses the formula on p. 112 of BJA3. Personally, I have always used the first of the two formulas on the top of page 113 for the formula. It is what was used to create all the charts that follow in the chapter. If you use the values you provided in that formula for risk, you will get 13.53%.
2. Obviously, for (2), there really is no time time constraint at all, so the answer, 13.45%, should indeed, be virtually the same as that for (1), with no time constraint.
3. No criticism intended, but I have no idea what the standard error of your simulations may be, or if they were done correctly, so I really can't comment except to say that you should also read pp. 140-146 to realize that while the formulas are reasonably accurate approximations, they are NOT perfect.
Don
Thanks Don, really appreciate your guidance and help.
Here are the rules of the above-mentioned game, there are 42 nos of balls( with label 1 to 42 ) in a box, player picks three numbers with total cost of $1, balls in box will be mixed "randomly" and 1 ball will be drawn from the box, if the draw number match any number of your pick, you will get pay $13.50( net win = 13.50 -1 = $12.50). However, from my few days observation, there are always 5( at least !) balls at certain corner of the box will never "mix" with other 37 balls !
Below is a simple program that how I model this game, could you(or any other experts) please help to find any mistakes or errors ?
42 Balls.jpg
"Don't Cast Your Pearls Before Swine" (Jesus)
I can't help you with programming. The e.v. and s.d. are correct. Why don't you run more than a million rounds? What is the standard error your way? In any event, what's the difference if the ROR is 12% or 13%? Why should you care? Just play the game with more than 144 units and make a fortune.
Don
My quick and dirty simulation agreed with your results and gave a ROR of 12.05% m/l. I chose a different stopping point to see if I could get substantially different results than yours. I also double checked your EV/SD and agree with your analysis.
I have encountered this sort of theory v. simulation discrepancy before, when analyzing loss rebates. The more that the higher moments in the distribution diverge from normal, the more theory diverges from simulations. Blackjack has a very tame distribution compared to this game.
Here is my code:
sim1.jpg
I edited this code and ran 10M trips and got:
10000000, 0.120483
Last edited by Eliot; 06-02-2020 at 11:04 AM.
Climate change blog: climatecasino.net
Eliot, by the way, I read your book, found I am “Fred” so I am using the time away from playing to strengthen up my game. As 21forme and others mentioned, my first 5 years were variance was with me and I came out ahead but 2019 and your book says the approach is all wrong. Thanks.
Bookmarks