Maximum Drawdown (Binomial tree)

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Discussion Overview

The discussion revolves around calculating the probability of a gambler experiencing a maximum drawdown, defined as losing more than or equal to a specified threshold of units, within a set number of trials. The context includes theoretical aspects of probability and random walks, particularly in relation to gambling scenarios.

Discussion Character

  • Exploratory
  • Technical explanation
  • Mathematical reasoning

Main Points Raised

  • One participant poses a problem regarding the probability of a gambler losing more than or equal to 10 units in 30 trials, given a winning probability of 0.6.
  • Another participant expresses confidence that the problem has been solved but does not provide details on the solution method.
  • A third participant inquires about the methods used to solve the problem, specifically asking whether generating functions or the Inclusion/Exclusion principle were employed.
  • A later reply introduces the reflection principle as a technique for counting random walk paths, suggesting it may simplify the problem-solving process.

Areas of Agreement / Disagreement

There is no consensus on the solution to the problem, as participants have not agreed on a specific method or outcome. Multiple approaches and techniques are mentioned, indicating a variety of perspectives on how to tackle the problem.

Contextual Notes

The discussion does not resolve the mathematical steps involved in calculating the probability, nor does it clarify the assumptions underlying the proposed methods.

kk007
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Imagine there is a game and a gambler has a prob. of P1 in winning one unit of capital in a trial and 1 - P1 in lossing one unit.

He wants to know the prob. of HAVING EVER lost more than or equal to a threshold no. of units (drawdown threshold) at or before the end of a number of trials.

For example, if his prob. winning 1 unit is 0.6 and lossing 1 unit is 0.4, what is his probability of having ever lost more than or equal to 10 units at or before the end of 30 trials?

To illustrate the "having ever" concept a bit more, imagine the gambler has 10 golden coins, at any time-step, he uses 1 golden coin for gambling, if he has lost all of them before the end of the 30 trials, his attempt is over.

Thanks in advance!
 
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Problem solved!

kk007 said:
Imagine there is a game and a gambler has a prob. of P1 in winning one unit of capital in a trial and 1 - P1 in lossing one unit.

He wants to know the prob. of HAVING EVER lost more than or equal to a threshold no. of units (drawdown threshold) at or before the end of a number of trials.

For example, if his prob. winning 1 unit is 0.6 and lossing 1 unit is 0.4, what is his probability of having ever lost more than or equal to 10 units at or before the end of 30 trials?

To illustrate the "having ever" concept a bit more, imagine the gambler has 10 golden coins, at any time-step, he uses 1 golden coin for gambling, if he has lost all of them before the end of the 30 trials, his attempt is over.

Thanks in advance!
 
Awesome. Did you solve it using a generating function or with the Inclusion/Exclusion principle? That seems like a hard problem.
 
I just now learned of the technique for counting random walk paths called the reflection principle, or method of images. It simplifies things a lot. Perhaps that's how you solved it so quickly.
 

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