*Rate of Ruin & Optimal N

#1
For a fair coin toss game, both prob(ruin) & prob(consecutive losses) increases with no. of trials.

Is the Rate of ruin .i.e.where prob(ruin) changes fastest, affected most by the number of consecutive losses?
If so, is there any condition or optimal N, maximum no. of consecutive losses to indicate a good time to exit the game?

In a bias coin toss game where prob(head)=0.7, what is the probability of 30 tails clustering (not necessary consecutively) in the first 50 tosses out of 100 trials?

Appreciate any suggestions.
 

BGM

TS Contributor
#2
Firstly, define what is "ruin" here - do you mean the number of tail (loss) observed is exceeding a certain number (the initial wealth)?

Secondly, do you know the probability of success/failure in each trial before you playing the game?

Thirdly, you need to define an objective function to maximize before finding the corresponding optimal stopping rule.
 
#3
Firstly, define what is "ruin" here - do you mean the number of tail (loss) observed is exceeding a certain number (the initial wealth)?

Secondly, do you know the probability of success/failure in each trial before you playing the game?

Thirdly, you need to define an objective function to maximize before finding the corresponding optimal stopping rule.
Yes, for fair coin toss of $1bet each toss, ruin means no. of tail exceeding Cstart(initial capital).

Prob=0.5 for head or tail

The objective is to find any condition or optimal N, no. of consecutive losses, where the prob(ruin) or rate of decline of capital is greatest/maximum, to exit the game.I am not sure what an objective function is but I guess that could be the formula for prob(ruin).
 

BGM

TS Contributor
#4
where the prob(ruin) or rate of decline of capital is greatest/maximum
Can you explain more?

For example, is it possible to play the game for infinite long time, or at most you can play a certain number of times? Or you must stop after your capital rise to a certain level?

Also for the probability of ruin in the infinite time case is a function of the current capital (and the probability of win/lose in each trial of course), which increase as the capital decrease; and I still not quite understand your objective.
 
#5
Can you explain more?
For example, is it possible to play the game for infinite long time, or at most you can play a certain number of times? Or you must stop after your capital rise to a certain level?
Also for the probability of ruin in the infinite time case is a function of the current capital (and the probability of win/lose in each trial of course), which increase as the capital decrease; and I still not quite understand your objective.
The fair coin toss game can be played infinitely. My goal is to find any mathematical expression that describes the optimal condition to quit, as a function of the variables N,no. of consecutive losses & Cnow,current balance$ for a fixed bet size,b. Since rate of prob(ruin) seems to be affected by N greatly, N should be equally important as Cnow.
 
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BGM

TS Contributor
#6
By assuming the independence of the trials, the ruin probability is independent of the past history.

Again, you want the optimal condition to quit - but optimal in what sense?
 
#7
By assuming the independence of the trials, the ruin probability is independent of the past history.
Again, you want the optimal condition to quit - but optimal in what sense?
When the no. of consecutive losses, N, has the greatest effect on the chances of current balance going to zero.Find a mathematical expression for this N in terms of Cnow & bet size, b (as % of Cnow).
 
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#8
To re-phrase,
When is the prob(streak of losses just enough to ruin its bankroll B) > 50%?
i.e. Is it possible to express this prob in terms of current bankroll B, b, N, p, q, etc?


Also, how do you get prob. of at least X tails in M tosses with prob. of tail = q, out of K trials
 

BGM

TS Contributor
#9
Sorry I do not understand how the ruin probability is related to the number of consecutive losses observed right now, assumed that each trial is independent. Or you mean anything else?

This just recall my first post in TS - a person also asking the question if there is a 10 consecutive losses, will the next trial has a higher probability to win...

Maybe I state one example:

You have $10 now and bet $1 each time. Two possible sample paths are

1. WWWWWLLLLL
2. WLWLWLWLWL

and both of them end up with $10 after 10 trials. And their ruin probability is the same after 10 trials.
 
#10
Also, how do you get prob. of at least X tails in M tosses with prob. of tail = q, out of K trials
Its ok, thanks. How about the above question?
For e.g,
In a bias coin toss game where prob(head)=0.7, what is the probability of 30 tails clustering (not necessary consecutively) in the first 50 tosses out of 100 trials?
 
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BGM

TS Contributor
#11
Not sure what do you mean by "clustering";

If you want to find what is the probability of 30 tails in the first 50 trials, then you can use the Binomial distribution to help; it is independent of the next 50 trials.
 
#12
Not sure what do you mean by "clustering";
If you want to find what is the probability of 30 tails in the first 50 trials, then you can use the Binomial distribution to help; it is independent of the next 50 trials.
How to use the Binomial distribution to help?
 

BGM

TS Contributor
#13
Assume each coin toss is independent and have the identical probability \( p \) to have a head. Then the number of heads in \( n \) trials \( \sim \text{Binomial}(n, p) \).

You can read more about the Binomial distribution in text book/wiki.