Suppose you wish to travel from the bottom left corner to the top right corner of an 8×8 grid by moving either one unit up or one unit right at a time. What is the probability that the path you take does not cross the grid’s diagonal?
Asked
Active
Viewed 237 times
0
-
The bottom left corner in part of the diagonal itself, isnt it ? – sentheta Feb 01 '20 at 07:03
-
@Vanwij yes . . . . – Feb 01 '20 at 07:13
1 Answers
1
It is twice the "Catalan number" $C_8$:
$2 C_8=2 \dfrac{1}{9}\binom{16}{8}$.
See this wonderfully illustrated article where you will find "$C_n$ is the number of monotonic lattice paths along the edges of a grid with n × n square cells, which do not pass above the diagonal".
(we have to count it twice because we must take into account paths that are above the diagonal).
A generalization : Number of paths that lie under the diagonal.
Jean Marie
- 81,803