Probability of Hitting Vertical vs Horizontal Lines in 2D Grid

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SUMMARY

The discussion centers on calculating the probability of hitting a positive vertical barrier (x) versus a negative horizontal barrier (y) in a 2D grid, starting from the origin (0,0). The user proposes that the probability of hitting -y is equivalent to hitting +x due to symmetry, and seeks validation or correction of this assumption. Additionally, the user requests guidance on proving this relationship and addressing scenarios where both barriers are positive.

PREREQUISITES
  • Understanding of probability theory, particularly in the context of random walks.
  • Familiarity with 2D grid models and their applications in stochastic processes.
  • Knowledge of symmetry principles in probability distributions.
  • Basic concepts of barriers in random walks and their implications on stopping times.
NEXT STEPS
  • Research the properties of random walks in 2D grids and their stopping probabilities.
  • Study the concept of symmetry in probability distributions and its applications.
  • Explore methods for proving equivalences in probability, particularly in stochastic processes.
  • Investigate scenarios involving multiple barriers and their effects on probability outcomes.
USEFUL FOR

This discussion is beneficial for mathematicians, statisticians, and researchers interested in stochastic processes, particularly those analyzing random walks and barrier probabilities in two-dimensional spaces.

iamback
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Hi, I have a problem

(1) where I need to compute the ratio of probabilities of hitting and stopping at a positive vertical barrier x vs hitting and stopping at a negative horizontal barrier y after starting from (0,0).

I feel that by symmetry, the answer to this would be the same as

(2) The probability of hitting -y vs hitting +x, horizontal lines in 2d grid,

which looks like being same as

(3) The probability of hitting -y vs x on a real line.

Can someone please tell me if my 1->2 assumption or 2->3 assumption is wrong. In which case, could someone please tell me how to proceed with the solution to 1.

However, if my assumption is right, can someone tell me how to proceed to prove it. Also, what would be a way to solve the case when both x and y are positive.



I shall be grateful for a response/hint/link.

Thanks.
PS: Please note that this is NOT a homework question and I really want to see this problem from a conceptual perspective. I've asked this question of many people, but none seem to be able to answer it.
 
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I think the issue is that (1) can be interpreted several ways. Could you provide more detail?
 

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