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.
Betting game
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?
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?
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.
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.
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!
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!