Path Counting - Chances of two people meeting?

In summary, the conversation is about a probability problem involving two individuals, Jeanine and Miguel, walking towards each other on a network of city streets. They both have a destination, but do not follow a specific route. The question asks for the probability of them meeting if they walk at the same rate. The conversation also mentions solving this problem for different grid sizes using Pascal's Triangle. The solution involves determining which corners can be reached in a certain number of moves, using Pascal's Triangle to find the number of ways to reach each corner, and then calculating the probability of each walker landing at a specific corner and adding them up. The answer in the book is 35/128.
  • #1
phono
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Homework Statement



I am having trouble with this problem.

A network of city streets forms square bloacks as shown in the diagram below.
http://img182.imageshack.us/my.php?image=librarypoolqs6.jpg

Jeanine leaves the library and walks toward the pool at the same time as Miguel leaves the pools and walks toward the lbrary. Neither person follows a particular route, except that both are always moving toward their destination. What is the probability that they will meet if they both walk at the same rate?

In addition, how would I solve this for a 1 by 1 grid, 2 by 2 grid, 3 by 3 grid,etc.?

I know that you have to use Pascal's Triangle and I think that they would have to meet on their "4th" moves. The answer in the book is 35/128 but I don't know how to get this.
 
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  • #2
First figure out which corners can be reached in 4 moves. Then figure out the number of ways to reach each of those corners (this is where Pascal's triangle comes in). Now figure out the probability that each walker will land at a given corner and add them up.
 

1. What is path counting and how is it related to the chances of two people meeting?

Path counting is a mathematical concept that involves counting the number of possible paths or routes between two points. It is related to the chances of two people meeting because it helps us calculate the likelihood of two people crossing paths based on the different routes they could take.

2. How is the concept of path counting used in real life?

Path counting can be used in various real-life scenarios, such as in transportation planning to determine the most efficient routes, in logistics to optimize delivery routes, and in epidemiology to study the spread of diseases.

3. What factors are considered in path counting for calculating the chances of two people meeting?

The factors that are considered in path counting for calculating the chances of two people meeting include the starting and ending points, the available paths or routes, the frequency of people using each path, and the probability of choosing a particular path.

4. How does the size of the population affect the chances of two people meeting?

The size of the population can significantly affect the chances of two people meeting. With a larger population, there are more paths and routes for people to take, increasing the chances of crossing paths. On the other hand, a smaller population may have fewer paths, making it less likely for two people to meet.

5. Are there any limitations to using path counting to calculate the chances of two people meeting?

Yes, there are some limitations to using path counting for calculating the chances of two people meeting. This method assumes that all paths are equally likely to be taken and does not consider other factors such as time of day, transportation modes, and individual preferences. Additionally, it may not account for chance encounters or unexpected events that could impact the likelihood of two people meeting.

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