You can determine the probabilities using the binomial distribution. Here is an old post I made on another website that will give you a start. It shows you how to calculate the probability of being ahead after a certain number of hours (or hands) of play. You will see that the chances of "beating the odds" using BS alone will decrease dramatically as you play for longer periods of time (or many sessions):
So let's look at the formula for determining your probability of
being ahead. The formula is fairly straightforward. You are trying to
find out what the likelihood of being in the upper (positive) part of
the bell curve is. As I showed in my last post, your results will
vary anywhere between several standard deviations. We want to find
out the probability that we will not be on the negative side of that
fluctuation.
Our hourly EV was 1.47 and our trip EV was 51.45. Even if we have a
negative swing of –50 units we will still end up with 1.47 units for
the trip, so a little negative fluctuation is okay. We want to make
sure that we end up with more than 0 units at the end of the trip. We
start by determining what percentage of the standard deviation our
Trip Win represents:
SDs = Square Root(Number of hours) * Hourly EV / Hourly SD
So for our example above, we get:
SDs = SQRT(35) * 1.47 / 25.21 = 5.92 * 0.058 = 0.345
So our Trip Win represents 0.345 standard deviations. There are
several ways to use this number to find exactly what we want to know.
THE HARD WAY:
Remember how I mentioned that a swing of one standard deviation
occurs about 68% of the time? Well, if you are familiar with
statistics you will be able to calculate that a swing of 0.345
standard deviations will occur about 36.32% of the time. That means
that you will lose money about 36% of the time. Since the rest of
those times you will be winning, you have a 64% of being ahead after
those 35 hours.
THE EASY BUT PAIN-IN-THE-ARSE WAY:
Look up the number 0.35 (rounded up) in the Cumulative Normal
Probability Distribution chart from any statistical book (I use the
one in Blackjack Attack). You will find that it points to 0.6368.
That gives you about a 64% chance of being ahead.
THE SIMPLE WAY:
Why didn't I just mention this way first?! I don't know. I guess I'm
just a mathematical masochist. =)
Instead of just giving away the formulas and saying "plug in your
numbers here" I like to explain how the formulas work as well. If you
can understand the reasoning behind the formulas you will understand
them much better. You may even see the game in a whole new light!
Anyway, here's the formula. Plug your numbers in here:
Go into an Excel spreadsheet and type =normsdist(0.345). Better yet,
type the whole formula in there!
=normsdist(sqrt(Number Of Hours) * Hourly EV / Hourly SD)
That way you can adjust the values to see the results from any number
of hours using any EV and SD. The spreadsheet will magically spit out
the number 0.634953… (depending on how many decimal places it shows).
If you convert that cell to a percentage (Format -> Cells ->
Percentage) it will move the decimal point and add that cute little
percent sign to tell you that you have about a 64% chance of being
ahead.
You can expand on this theory to find the probability of being ahead by $X after a given amount of time and the propbability of losing $Y in that same time. The results above are for a player with an advantage, but the formula will work for a BS player as well.
-Sonny-
