SOLUTION: Let A(2,1) and B(8,5) be the two points in the plane that are the nodes of a grid described in the module. If all paths are equally likely, what would be the probability of choosin

Algebra ->  Probability-and-statistics -> SOLUTION: Let A(2,1) and B(8,5) be the two points in the plane that are the nodes of a grid described in the module. If all paths are equally likely, what would be the probability of choosin      Log On


   

Question 1129609: Let A(2,1) and B(8,5) be the two points in the plane that are the nodes of a grid described in the module. If all paths are equally likely, what would be the probability of choosing a path that begins with an up move and ends with an up move? (each path is made up of moves that either go up one unit or over one unit to the right).
Answer by greenestamps(13367) About Me  (Show Source):
You can put this solution on YOUR website!


A path from (2,1) to (8,5) must go 6 moves to the right ("RRRRRR") and 4 moves up ("UUUU"). The total number of paths is the number of arrangements of "RRRRRRUUUU"; by a well-known counting principle, that number of ways is

10%21%2F%28%286%21%29%284%21%29%29+=+C%2810%2C4%29+=+210

If a path must begin and end with a move up, then the number of paths is the number of paths from (2,2) to (8,4), which is

8%21%2F%28%286%21%29%282%21%29%29+=+C%288%2C2%29+=+28

So the probability that a path from (2,1) to (8,5) begins and ends with an up move is

28%2F210+=+2%2F15