# Thread: Path Counting - Chances of two people meeting?

1. ## Path Counting - Chances of two people meeting?

I am having trouble with this problem.

A network of city streets forms square bloacks as shown in the diagram below.
ImageShack - Hosting :: 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 the answer in the book is 35/128 but I don't know how to get this.

2. Look carefully at the diagram you supplied.
Did you mean to have a 4x4. You put a 4x3.
This problem is usually done on a square grid.

3. Yea sorry I messed it up, I fixed it.

4. Look at my diagram. If the two will meet that will happen at one of the points on the diagonal.
Here are the number of paths from either the pool or the library to the diagonal $\left( {\begin{array}{*{20}c}
A & B & C & D & E \\
1 & 4 & 6 & 4 & 1 \\
\end{array}} \right)$

You should recognize those numbers.
The probability that they will meet at point C is $\left( {\frac{6}{{16}}} \right)\left( {\frac{6}{{16}}} \right)$.

Now you calculate the probabilities of their meeting at each of the other points on the diagonal and add them up. If done correctly the answer will be $\frac {35}{128}$.

Also, I have given you a pattern to follow for the other cases and a generalization.