Probability of Hitting Vertical vs Horizontal Lines in 2D Grid

In summary, the speaker is asking for help with a problem involving computing the ratio of probabilities for hitting and stopping at a positive vertical barrier and a negative horizontal barrier after starting from (0,0). They believe that the answer may be the same as the probability of hitting a negative horizontal barrier versus a positive vertical barrier in a 2d grid, or hitting a negative horizontal barrier versus a positive vertical barrier on a real line. They are seeking clarification on their assumptions and how to proceed with solving the problem. They also clarify that this is not a homework question and they are looking for a conceptual perspective.
  • #1
iamback
1
0
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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  • #2


I think the issue is that (1) can be interpreted several ways. Could you provide more detail?
 

1. What is the probability of hitting a horizontal line on a 2D grid?

The probability of hitting a horizontal line on a 2D grid is 1 in 2, or 50%. This is because there are an equal number of horizontal and vertical lines on a 2D grid, and each line has an equal chance of being hit.

2. How does the number of horizontal lines affect the probability of hitting one on a 2D grid?

The number of horizontal lines does not affect the probability of hitting one on a 2D grid. As long as there is at least one horizontal line on the grid, the probability remains at 50%. The number of horizontal lines only affects the likelihood of hitting a specific line, not any horizontal line in general.

3. Is the probability of hitting a vertical line different from the probability of hitting a horizontal line on a 2D grid?

No, the probability of hitting a vertical line is the same as the probability of hitting a horizontal line on a 2D grid. Both have a probability of 1 in 2, or 50%. This is because the grid is symmetrical and each line has an equal chance of being hit.

4. How does the size of the grid affect the probability of hitting a vertical or horizontal line?

The size of the grid does not affect the probability of hitting a vertical or horizontal line. As long as the grid remains symmetrical, the probability remains at 1 in 2, or 50%. However, as the size of the grid increases, the number of lines also increases, which may affect the likelihood of hitting a specific line.

5. Does the position of the starting point affect the probability of hitting a vertical or horizontal line on a 2D grid?

No, the position of the starting point does not affect the probability of hitting a vertical or horizontal line on a 2D grid. As long as the starting point is within the grid boundaries, the probability remains at 1 in 2, or 50%. The starting point only affects the likelihood of hitting a specific line, not any vertical or horizontal line in general.

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