Statistician/Mathimagition needed.

[longwolf]

I'm working on a Blackjack Strategies tester (in VB6).
One of the calculations I would like it to give is called Risk of Ruin or ROR.
ROR tells you the percentage chance of losing your whole bank roll before, say, doubling it.

Here's an example of some figures that might come up.
Bank Roll: 3000
Win/Loss Ratio: 96.84% chance of winning each session, or about 30 wins to 1 loss
Average Win Per winning session: $53.33
Average Loss Per losing session: $1,297.97

So can someone think of a formula to give the ROR?

[Rassis]

I don’t know anything about that specific game and its rules. All I know is that Risk is calculated multiplying the chance of something undesirable happens by its consequences (dollars, individuals injured, dead, or other).

From the numbers that you give, I only can deduce that the expected value of any outcome is favourable to the player and equal to:

($53.33 x 30 – $1,297.77 x 1) / (30 + 1) = $9.75

But I suspect that this is not the kind of answer you are aiming at…

By the way, should "Bank Roll = 3000" play any role in the calculus?

[longwolf]

Hi Rassis,

The formula you're giving tells the average profit per session.
In this case the Black Jack system used will win in the long run.

The Bank is the point of what I'm looking for now.
The odds may be 30 to 1 in favor of a single winning session, but the ROR calculation would be for multiple sessions.
All winning sessions get added to the Bank, while losing sessions are subtracted.

For example, you might win 90 sessions in a row then lose 3 sessions.
In which case you'd be in great shape.

Or you might start by loosing 3 sessions and then have 90 winning ones.
But if you lost the 3 first your Bank would be gone and you would be RUINED.
i.e. you wouldn't have the money to play the other 90 sessions.

Now I think the odds of losing the first 3 sessions in a row would be something like 30 * 30 * 30 to 1, or 2700 to 1 against.

But a more likely loosing scenario would be something like;
Win 5, lose 1, win 4, lose 1, win 6, lose 1
Or it could go on much longer than that.

Bank, Win/Loss ratio, Ave. Win and Ave. Loss are all variable depending on the system being tested.
You can reduce your ROR by increasing your Bank, Win/Loss ratio or your Ave. Win.
You can also reduce the ROR by reducing your Ave. Loss.

Now let's say you've found two winning systems.
One system has a long term average win of $5 per session, but the other makes an average of $60 per session.

You'd pick the $60 system right?
But if the $5 system had a ROR of 2% while the $60 system had a ROR of 45% you'd be better off playing the $5 one unless you can raise a big enough Bank Roll to bring the $60 systems ROR down to an acceptable value.

So, again, can someone think of a formula to find the chance of going broke using the variables Bank, Win/Loss ratio, Ave. Win and Ave. Loss?

[Rassis]

Longwolf,

There is one more variable, which is important, that you are not accounting for: it is the number of sessions before going broke.

I don’t know how to work out a formula that gives the probability of going broke. Perhaps some body else in this forum can give you an answer on this. All I can do is to show you how to deal with such a problem using Monte-Carlo simulation.

Please find an Excel file attached where the Black Jack game is programmed.

You can give any values to the variables (blue cells): Bank roll, Average win, Average loss and the Win/Loss ratio. By pressing the function key F9 you will notice that numbers in columns F, G and H change and cell J9 will return the number of sessions necessary for the bank roll to become negative. Then you enter the sheet “Repeater” and repeat the outcome in cell J9 so many times as you wish (a few hundreds perhaps will do), in order to get statistical significance, by pressing the button “Press to calculate”. In the end, see the sheet “Histogram” where the frequency distribution of number of sessions to go bankrupt is displayed. In this sheet you get the answers for your question.

For instance, what is the risk of ruin (ROR) using the data in sheet “Data”:

- until session 45? The answer is in column I and is 0,84;
- until session 30? The answer is given in cell L7 (0,58) by entering 30 in cell L5;
- until session 4? The answer is given in cell L7 (0,00) by entering 4 in cell L5;
- until session 136? The answer is given in cell L7 (1,00) by entering 136 in cell L5;
- until session 150? The answer is given in cell L7 (1,00) by entering 150 in cell L5;
- and so forth…

Or in reverse order:

- how many sessions are expected to be safe for a ROR = 0,4 (before going broke)? The answer is given in cell L10 (22,6 or 22 sessions) by entering 0,4 in cell L9;
- how many sessions are expected to be absolutely safe (for a ROR = 0,0)? The answer is given in cell L10 (6) by entering 0 in cell L9;
- What is the minimal number of sessions to be absolutely certain to go broke (ROR = 1)? The answer is given in cell L10 (136 sessions) by entering 1 in cell L9.

I hope this helps.