In Problem 10, assume that if a sensor has a choice, it will send the message to the above sensor with probability p_a and will send the message to the sensor to the right with probability p_r = 1 −p_a. What is the probability that the sensor located at point (10, 5) receives the message?

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)