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## #1 2013-02-12 09:57:54

nflguy
Member
Registered: 2013-01-31
Posts: 18

### Calculating Rate of Return on Investment

Say you went to the race track every day and bet on 3 horse races.  You are very good at what you do and are able to pick the correct horse 55% of the time.  The amount you wager on each race is 5% of your current bankroll which you calculate after each race.  Each time you win a race by picking the correct horse you add 5% to your bankroll.  Each time you are incorrect you lose 5.5% of your bankroll.

The question is if you have a starting bankroll of $5000 how long would it take to turn that intial$5000 into 1 million dollars and how do I figure that out.

Second question is do the order of wins and lossess matter.

nflguy

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## #2 2013-02-12 15:18:29

bobbym
bumpkin
From: Bumpkinland
Registered: 2009-04-12
Posts: 109,606

### Re: Calculating Rate of Return on Investment

Hi nflguy;

There is a chance of going broke. What happens then? Does the player get more money or is the game over?

Second question is do the order of wins and lossess matter.

I do not believe that the order counts.

Last edited by bobbym (2013-02-12 15:40:02)

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

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## #3 2013-02-12 15:50:52

nflguy
Member
Registered: 2013-01-31
Posts: 18

### Re: Calculating Rate of Return on Investment

You would keep adjusting your bankroll after each result so there would be no way to go broke.

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## #4 2013-02-12 15:59:29

bobbym
bumpkin
From: Bumpkinland
Registered: 2009-04-12
Posts: 109,606

### Re: Calculating Rate of Return on Investment

That answer is fine if we were discussing continuous values. But we are talking about discrete values. There is a smallest unit of currency. After 232 straight losses you would have less than a penny. Essentially you are broke.

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

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## #5 2013-02-12 16:01:48

nflguy
Member
Registered: 2013-01-31
Posts: 18

### Re: Calculating Rate of Return on Investment

but isn't 232 losses virtually impossible, it is worse odds than a coin landing heads 232 times in a row.

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## #6 2013-02-12 16:06:47

bobbym
bumpkin
From: Bumpkinland
Registered: 2009-04-12
Posts: 109,606

### Re: Calculating Rate of Return on Investment

Virtually impossible and really impossible are not the same thing. It does not have to be 232 in a row. It just has to be 232 more losers than winners. Yes, it is a longshot but it changes the problem.

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

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## #7 2013-02-12 16:10:54

nflguy
Member
Registered: 2013-01-31
Posts: 18

Ok so if you have a starting bankroll of $5000 how long would it take to turn that intial$5000 into 1 million dollars AND what are the chances you end up going bankrupt if you dont reach the goal.  OR you can say that if you go bankrupt you are allowed to replinish the starting $5000 again. Offline ## #8 2013-02-12 16:16:03 bobbym bumpkin From: Bumpkinland Registered: 2009-04-12 Posts: 109,606 ### Re: Calculating Rate of Return on Investment Hi; You have to be careful how you a phrase a problem. The smallest difference in rules can turn a solvable problem into one that can not be solved. With the replenish rule you are taking the problem out of the realm of mathematics and into the realm of programming. With the going broke-game over option there might be a solution using Markov chains. This is a little optimistic at this point but I am hoping. In mathematics, you don't understand things. You just get used to them. If it ain't broke, fix it until it is. Always satisfy the Prime Directive of getting the right answer above all else. Offline ## #9 2013-02-12 16:26:05 nflguy Member Registered: 2013-01-31 Posts: 18 ### Re: Calculating Rate of Return on Investment Ok then the going broke option it is. I have heard of Markov chains and monte carlo but I honestly have no idea how to input the data into them for this problem. Offline ## #10 2013-02-12 16:38:35 bobbym bumpkin From: Bumpkinland Registered: 2009-04-12 Posts: 109,606 ### Re: Calculating Rate of Return on Investment Hi;$5000 into 1 million dollars AND what are the chances you end up going bankrupt if you dont reach the goal.

One more thing now. "Chances," refers to a probability, but your original question wanted how long. Which do you require?

Also, you want the loss to cost 5.5% and the win to only gain 5% ?

Last edited by bobbym (2013-02-13 10:03:39)

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

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## #11 2013-02-14 07:39:34

bobbym
bumpkin
From: Bumpkinland
Registered: 2009-04-12
Posts: 109,606

### Re: Calculating Rate of Return on Investment

Hi;

I have gone with your original idea that you can not run out of money because it simplified the problem.

I am getting an expected number of 1930 plays.

$\large \inline \bg_green \fn_jvn a(n+1)=a(n)+(0.55 \cdot 0.05-0.45 \cdot .055)a(n)$

$\large \inline \bg_green \fn_jvn a(n+1)=1.00275 a(n)$

Last edited by bobbym (2013-02-14 07:55:59)

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

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## #12 2013-02-14 08:23:33

nflguy
Member
Registered: 2013-01-31
Posts: 18

### Re: Calculating Rate of Return on Investment

Thanks Bobby can you teach me how you did this?

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## #13 2013-02-14 08:27:43

bobbym
bumpkin
From: Bumpkinland
Registered: 2009-04-12
Posts: 109,606

### Re: Calculating Rate of Return on Investment

Hi;

I am not very confident in the answer myself.
It will take another week before I am confident in the result. So far, all I can say is it matches some simulations I did.

It uses a recurrence relation.

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

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## #14 2013-02-14 08:29:13

nflguy
Member
Registered: 2013-01-31
Posts: 18

### Re: Calculating Rate of Return on Investment

ok in the meantime I will read up on recurrence relation

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## #15 2013-02-14 08:30:46

bobbym
bumpkin
From: Bumpkinland
Registered: 2009-04-12
Posts: 109,606

### Re: Calculating Rate of Return on Investment

Look it up on google and stay clear of wikipedia. They are no good unless you already know the subject.

Come back if you have any question on a recurrence or difference equation as they are also called.

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

Offline