## Betting game

Decision and strategy, simulations and probability models, card and board games, ...
Alhazen
Posts: 94
Joined: Sun Feb 20, 2011 9:55 pm

### Betting game

Betting game

An urn contains 10 balls :
- 6 are black
- 4 are white
The picking is made randomly from the urn : one ball at time without replacement.
So we expect a total of 10 drawings.

Now the betting conditions :
- you bet before each drawing
- you have to bet in each drawing at least 1 dollar
- you have to bet a black ball in each drawing
- if the result is a black ball you get paid 1.30 dollars for one dollar bet. So your profit will be 30 cents.
- if the result is a white ball you loose you bet
- you can bet any amount of money (no limit)
- you have a lot of money but you need to use the minimum (k dollars) such as your capital at the end of the 10 drawings will exceed 1.20 k dollars.

So the core of the problem is how much money you will bet in each of the ten drawings.

thundre
Posts: 356
Joined: Sun Mar 27, 2011 9:01 am

### Re: Betting game

If the reward is only .30 for a risk of 1, the game is only favorable if black:white > 3.33333:1. There is a 60% chance that this will be the case for the last ball (1:0), and lower before that.

last 1: 60%
last 2: 33.33%
last 3: 16.66%
last 4: 7.14%
last 5: 2.38% 5:0, 23.8% 4:1

Of course, it the last 4 are all black, you know ahead of time that the last 3, 2, 1 will also be sure wins.

I don't quite understand the question you're asking. Are you trying to bet a minimum expected amount for a guaranteed profit, or bet a minimum worst-case amount for an expected profit, or something else?

Alhazen
Posts: 94
Joined: Sun Feb 20, 2011 9:55 pm

### Re: Betting game

What is required is that (I quote you) you " bet a minimum expected amount for a guaranteed profit").
You can reuse your wins.
So if your capital required to survive all the cases is 600 dollars for example you have to guaranty a profit of at least 600*0.20=120 dollars.
For example with 600 dollars which is the minimal capital required you have to face any event. So you have to imagine the worst case.
2^10 =1024 cases to analyze.
We can do that by starting to compute what is required in each drawing : the minimal capital required will grow quickly.
With 1.30 to 1 you can use a martingale 1 10 100 1000 10000 .... and you are sure to win. It will be then easy.

Alhazen
Posts: 94
Joined: Sun Feb 20, 2011 9:55 pm

### Re: Betting game

It is easy too to guaranty 1.20 of the capital used to bet.
144.597 dollars and it is done no matter what happens. You always guaranty a profit of at least 20%.
But 1.20 of the minimal capital required is a big headache!