In Problem 10, assume that all the appropriate paths are equally likely. What is the probability that the sensor located at point (10, 5) receives the message? That is, what is the probability that a randomly chosen path from (0, 0) to (20, 10) goes through the point (10, 5)?

Problem 10

A wireless sensor grid consists of 21 × 11 = 231 sensor nodes that are located at points (i, j) in the plane such that i ∈ {0, 1,⋯, 20} and j ∈ {0, 1, 2,⋯, 10} as shown in Figure 2.1. The sensor node located at point (0, 0) needs to send a message to a node located at (20, 10). The messages are sent to the destination by going from each sensor to a neighboring sensor located above or to the right. That is, we assume that each node located at point (i, j) will only send messages to the nodes located at (i + 1, j) or (i, j + 1). How many different paths do exist for sending the message from node (0, 0) to node (20, 10)?

(0,10). (0.9). (0,2) (0,1). (1.10) (2.10) (0,0) (1.0) (2.0) (3,0) (19,10) - Fig.2.1 Figure for Problem 10. (20,10) (20.9) (20.1) (19,0) (20,0)