Krdytkyu

I have a mathematical problem that I am trying to solve but are not sure how to do it. It is quite complex but I will try to describe the exact problem. The last 5 lines in this description of the problem is what the question of the problem boils down to.

Link to image that shows all the statistics in excel needed for below problem:
Image of the statistics

Description of mathematical problem:
1. On average we want to invest 100 dollar in Total through all betlevels. This can mean that we sometimes invest 150 in total and sometimes only 50. But on average this will be about 100 over time (Law of Large Numbers)

2. A person will make bets on a table. EACH bet have a sequence of 1-4 bets (which are the betlevels 1-4). The trick is that sometimes the sequence will only be betlevel 1 sometimes it will be 1 and 2, sometimes 1,2,3 and sometimes all 4. We never know beforehand! As we can see 1372 of Total 3430 bets will Stop on level 1. 1029 of the bets will Reach betlevel 2 and will then bet on BOTH level 1 and level 2. This is how the logic goes down to betlevel 4.




3. Now comes the problem. What is the OPTIMAL way(I stress OPTIMAL) to invest through the 4 betlevels. I have made 2 theoretical examples(see above image) where I take as an example for level 2: (33.33 x 1029 bets x 0.23% = 64.02 profit)

Example 1) On average bet 33.33 on level 2,3,4 which gives: 182.91 in Total Profit. (Level 1 invest 0 as this levels shows -0.15%)

Example 2) On average bet 33.33 on level 2. 36.33 on level 3. 30.33 on level 4 which gives: 185.07 in Total Profit. (Level 1 invest 0 as this levels shows -0.15%)

This shows that example 2 is a more OPTIMAL way to invest.
Remember we want in our Equation on average invest 100 through betlevel 1-4.

The problem boils down to this Question:
We need to invest a sum on RANDOM on EACH betlevel using some kind of equation using the statistics to reach the OPTIMAL Total Profit Result over time. (Law of Large Numbers). My question is how this equation RANDOM function/equation will look like and how/what dollar amount we will invest on each level? (I think the green fields in the above image are the important information but feel free to use the other information too)

Many Thanks!

It is better for me to put a much more simple example. I will begin with my effort for a solution(like Jeopardy). After that comes all info and statistics needed in order to find a solution.



The approach I have used (using the statistics)
Those 2 lines are not completely correct and is the complete solution I am looking for:
For Betlevel 1: Random generator with 50% chance bet: 100d or 59.23d
For Betlevel 2: Random generator with 50% chance bet: 59.23d or 40.77d

That gives me those results:
A) Original as seen in below description is a total of 34800d with 100d invested for each bet which is my goal to strive for with this mathematical problem
B) My approach/effort to a mathematical solution gives 31424d with 88d invested for each bet wich is about 90% accurate.

My question simply is what I am doing wrong. How will I do this to be as close as possible to the original result 34800d ?

See image or the statistics below that I work with

                      Avg%/Bet                Nr Bets REACHING               Avg% *                    Accumulated%

                                                        this Level                                                                   this Level

Level 1:        0.23%                      100000 - (83.33%)                  23000%                27600%

Level 2:       0.95%                      20000 - (16.67%)                     19000%                19000%
                                                       Total Bets: 120000                                                 Total%: 46600%


Information/statistics for mathematical problem:

1. We will do a Total of 120000 bets

2. 100000 of the bets will reach/stop on betlevel 1

3. 20000 of the bets will reach/stop on betlevel 2

(This means that those 20000 bets has been on betlevel 1)

4. Total % for bets that stopped on Level 1: 0.23% * 100000 = 23000%

5. Total % for bets that stopped on Level 2: 0.95% * 20000 = 19000%

6. Total % for bets that HAS BEEN on Level 1 : 23000% + (0.23 * 20000) = 27600%

7. Total % for bets that HAS BEEN on Level 2 : 19000%

8. Total % Accumulated: 46600%

9. Level1 stands for: 27600%/46600% = 59.23% of all profit

10. Level2 stands for: 19000%/46600% = 40.77% of all profit


Now is the Goal is to put 100d in each bet that can consists of 1 -OR- 2 levels of bets like this:


(Goal is to put 100$ in each bet. Line 1 reach level 1. Line 2,3 is a bet that reach level 2)

Level 1: 100000 * 100d * 0.0023 = 23000d
Level 1: 20000 * 50d * 0.0023 = 2300d

Level 2: 20000 * 50d * 0.0095 = 9500d

                                                             Total: 34800d



The mathematical problem is now this:

We never know if the bet we start with at level 1 will continue to LEVEL 2 or stop at LEVEL 1. This is the WHOLE problem. So we must now bet an unknown amount on LEVEL 1 and LEVEL 2. The goal is on average bet as close to 100d as possible for each bet.