A binary message (either 0 or 1) is transmitted by wire from location A to B. However, the data sent over the wire is subject to random disturbance. In order to reduce possible errors, the value 2 is sent over the wire when the message is 1, and the value -2 is sent when the message is 0. If x (x = ±2) is the value sent at location A, then the value R received at location B is given by R = x + N, where N is the channel noise disturbance which follows a standard normal distribution. When the message is received at location B, the receiver decodes it according to the following rule: if R > 0.5, then a message of 1 is concluded, and if R < 0.5, then a message of 0 is concluded. (i) Suppose that a random message that is equally likely to be 0 or 1 is sent to location B. What is the probability that the receiver decodes the message incorrectly? (ii) Suppose that a string of 1, 0, 0, 1 is sent to location B sequentially, and the channel disturbance acts on the 4 binary messages independently. What is the probability that the receiver decodes the information precisely? (iii) Suppose that a string consisting of 100 independent random binary messages is sent to location B sequentially, and the channel disturbance acts on the messages independently. The original information will be lost if the receiver makes more than 2 errors when decoding the messages. Use Poisson approximation to compute the probability that the information is preserved after the transmission.