How to calculate RoR?

#1
If I assume a normal distribution, I know how to calculate the probability of being below a certain point after n number of samples.

But how do you calculate the chance of ever being below a certain point after n or less samples?

I might play 200 hands in a night and know there's a 10% chance I'll end up down 50 units, but what's the chance that I'll be down 50 units at any point during the night?
 

Southpaw

Well-Known Member
#4
r=N((B-ev)/sd) + e^((2*ev*B)/var) * N((B+ev)/sd)

where

N = the cumulative normal distribution function of what is found in parenthesis (basically, you obtain an integer inside parenthesis and then reference a table to find what value of N corresponds to that integer). Use this table here to find what value of N corresponds to the integer inside the parenthesis: http://www.cs.washington.edu/homes/jrl/normal_cdf.pdf

e = the base of the natural logarithm system (~2.7183)

B = Bankroll (expressed as a negative number!)

EV = Expected value or win

sd = Standard deviation from the win or ev

var = variance of the win (the square of the the sd) in units^2

Let me know if you have any questions about the formula. It is a tricky one.

Spaw
 
#5
Southpaw said:
EV = Expected value or win

sd = Standard deviation from the win or ev

var = variance of the win (the square of the the sd) in units^2
How would you calculate these since they change depending on where you are in the deck, what cards have come out, etc.?
 

Southpaw

Well-Known Member
#6
bobstaman said:
How would you calculate these since they change depending on where you are in the deck, what cards have come out, etc.?
You are not concerned with what the EV or SD is at any given point at the shoe; rather, you would run a simulation or do some math and figure out what the overall EV and SD would be for the whole trip. (Note that this is the Trip RoR formula meaning that you are calculating the RoR assuming a given period of time. This contrasts with simple RoR that assumes an infinite number of hands.)

Spaw
 
#7
Southpaw said:
r=N((B-ev)/sd) + e^((2*ev*B)/var) * N((B+ev)/sd)

where

N = the cumulative normal distribution function of what is found in parenthesis (basically, you obtain an integer inside parenthesis and then reference a table to find what value of N corresponds to that integer). Use this table here to find what value of N corresponds to the integer inside the parenthesis: http://www.cs.washington.edu/homes/jrl/normal_cdf.pdf

e = the base of the natural logarithm system (~2.7183)

B = Bankroll (expressed as a negative number!)

EV = Expected value or win

sd = Standard deviation from the win or ev

var = variance of the win (the square of the the sd) in units^2

Let me know if you have any questions about the formula. It is a tricky one.

Spaw
Thanks, Southpaw.

It looks ok under typical situations. But if I play a trillion $1 hands with a bankroll of $1, the formula says my RoR is virtually zero, but common sense says that I'm very likely to quickly lose my teeny bankroll.

I was expecting to see an input for the number of hands.

Or does the formula only apply to reasonable questions like, "What kind of bankroll do I need for my next trip, which might be between one and 40 hours?" and I should leave it at that.

I'm sorry if I seem picky - I just want to make sure I use the formula correctly.
 

Southpaw

Well-Known Member
#8
Mr. Ed,

This formula is one that determines the RoR after a given number of samples just as you had indicated in your OP.

Indeed, there is no variable input immediately visible for time. However, time, or the number of hands played is built into the input for EV and SD.

Suppose you wanted to determine your RoR, given your bank and that you'll be playing for "t" hours. You need to calculate the EV after "t" hours first. (EV/hour)*(t hours). You do not input the hourly EV into the equation, but rather the EV after t hours.

Similarly, you need to calculate the standard-deviation after "t" hours. Recall that standard-deviation after "t" hours equals (hourly standard-deviation)(sqrt(t)). Again, you are not entering hourly standard-deviation into the equation, but rather you are entering the standard-deviation after "t" hours.

Hope this helps,

Spaw

P.S.--I am not certain I understand what you are saying with your case of the $1 bankroll and 1 trillion hands of $1 blackjack.
 
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