Show all work and answer please, thank you

Problem 1. (12 points) A communications channel transmits the digits 0 and 1. However, due to static, the probability that any transmitted digit will be incorrectly received is 0.2. To reduce the chance of error, we transmit 0000 instead of 0 and 1111 instead of 1. Suppose that we want to transmit an important message consisting of the binary digit 1 and suppose the probabilities of receiving different digits incor- rectly among the 4 transmitted digits are independent, if the receiver of the message garuses: majority decoding, what is the probability that the message will be incorrectly ' received? (By majority decoding we mean that the message is decoded as \"0\" if there r areiat least three zeros in the message received, and as \"1\" if there are at least three _ '_ - r'ones in the message received.)