problem 267

[hisoka-san]

Say after 100 bets I have 1e9 pounds. Can I quit and take the money?

[stijn263]

Problem 267 (View Problem) says:

you can choose a fixed proportion, f, of your capital to bet on a fair coin toss repeatedly for 1000 tosses.

So, no you can not. You bet the same fixed proportion of your capital each time. Good luck

[hisoka-san]

what I didn't take into account that in case of win, bet amount is returned. Thanks

[blazeiliev]

I'm troubled with the text of problem 267. It says

"...For example, if f = 1/4, for the first toss you bet £0.25, and if heads comes up you win £0.5 and so then have £1.5. You then bet £0.375 and..."

How come from having £1 you end up having £1.5? If you bet £0.25 and win, you should have £1.25.

[Lord_Farin]

I'm troubled with the text of problem 267. It says

"...For example, if f = 1/4, for the first toss you bet £0.25, and if heads comes up you win £0.5 and so then have £1.5. You then bet £0.375 and..."

How come from having £1 you end up having £1.5? If you bet £0.25 and win, you should have £1.25.

Win is measured respective to the amount before betting, not that during betting. In other words, the win excludes the return of the bet.

[blazeiliev]

I'm troubled with the text of problem 267. It says

"...For example, if f = 1/4, for the first toss you bet £0.25, and if heads comes up you win £0.5 and so then have £1.5. You then bet £0.375 and..."

How come from having £1 you end up having £1.5? If you bet £0.25 and win, you should have £1.25.

Win is measured respective to the amount before betting, not that during betting. In other words, the win excludes the return of the bet.

I'm still having trouble understanding. Can you give me an example with numbers.

[hk]

First of all you choose a fraction f, say f=0.25.
You start having 1 pound.

So your first bet is 0.25*1=0.25.
Then we have the rule:

Your return is double your bet for heads and you lose your bet for tails.

So if you have heads your return is twice your bet=2*0.25=0.5, this is added to what you have making 1.5.

At the second toss your bet is 0.25*1.5=0.375.
If you now have tails your bet is subtracted from what you have: 1.5-0.375=1.125

[Lord_Farin]

I'm still having trouble understanding. Can you give me an example with numbers.

Suppose we can bet £1 for a 50% chance to win £2. Then:
if we lose, we get £0
if we win, we get £3 = £1 for the return of the bet, and £2 of the win.
In other words, 'win' is synonymous to 'profit.' Or phrased differently again, 'win' = 'money after bet' - 'money before bet,' presumed this is not negative.

You can also view this as 'virtual betting' where you say you will bet £1, and if you lose, you will receive a bill for £1 representing the loss of your bet. If you win, you will get a payment of £2 (thus then you have £3).

[blazeiliev]

Thanks, I think I finally got it.
You win twice your investment plus your initial bet (if you win)
and you lose only your bet (if you lose).
Makes sense to be a unique investment opportunity

[tobiaso]

But do you have to choose your own f? Or is the given f already 1/4?
Or is the question what is the best f you can choose?

And i assume this is random, how can we know the random generator of a specific language is the same as any other one?
All throws have a random 50% chance of winning or losing, right?

[Lord_Farin]

But do you have to choose your own f? Or is the given f already 1/4?
Or is the question what is the best f you can choose?

And i assume this is random, how can we know the random generator of a specific language is the same as any other one?
All throws have a random 50% chance of winning or losing, right?

The fraction f is uniquely determined by having the stated maximality property. Using it, calculate the desired probability.
The randomness is assumed to be mathematical randomness. This is different from the pseudorandomness any computable algorithm gives. You might want to look up some information about statistics.

As a response to your last question: This would mean that the probability of throwing a 1 or not with a fair die are both 50%. You will probably agree such is not the case... But again, look up some statistics.

[thundre]

The fraction f is uniquely determined by having the stated maximality property.

False. f is a real number, and there is an interval which yields the maximum probability.

All throws have a random 50% chance of winning or losing, right?

Right.

[Lord_Farin]

I apologise for posting blatant lies. I should read the problem description more carefully the next time somebody asks a question.

[tommyjcarpenter]

Is the smallest denomination possible considered to be .01? Meaning, if f is .5, and you have .01 remaining, you bet and lose, does the iteration continue with having .005 of money?

If not, what is the smallest denomination of money that is considered > 0?

[Animus]

The iteration continues with any amout of money, no rounding of the amount left (or gained) happens during the process.

[walgong]

I have a problem concerning this "maximize your chances".
Usually there is the notion of expectation value. I guess it does not apply here, as then in my opinion we would have f=1.
But what is then to be maximized? Is it the number of the different possible outcomes of the 1000 tosses where we get more than 1000000000?

[Circling]

I have a problem concerning this "maximize your chances".
Usually there is the notion of expectation value. I guess it does not apply here, as then in my opinion we would have f=1.
But what is then to be maximized? Is it the number of the different possible outcomes of the 1000 tosses where we get more than 1000000000?

If f was 1, all your savings would be gone in a toss that comes as tails.
The chance of tails never occcuring is 1 / (2 ** 1000)

So no, f = 1 doesn't maximize the chance. In fact it actually minimizes it.

[shaiephraim]

I'm troubled with the text of problem 267. It says

"...For example, if f = 1/4, for the first toss you bet £0.25, and if heads comes up you win £0.5 and so then have £1.5. You then bet £0.375 and..."

How come from having £1 you end up having £1.5? If you bet £0.25 and win, you should have £1.25.

I had the same issue with the wording.. I think it should be clarified more clearly