**a.** The telemetry system of a certain spacecraft has a probability of 1 percent of transmitting an erroneous bit. One way to increase data reliability would be to repeat each message bit three times. For example, . . . 010110 . . . would become . . . 000111000111111000 . . . , if no errors occur. If it is decided to interpret any of the triplets 000, 001, 010, or 100 as O and any of the triplets 01 1 , 101, 110, or 111 as 1, find the probability of error in the interpretation of a message bit, assuming the transmission of each bit is independent.

We see that we can reduce the probability of a transmission error in a single bit from 1 percent to 0.03 percent, but at a cost of sending three times as many bits as are actually needed for the message. To put it a different way, the desired message would be sent one-third as quickly.

**b. **More efficient error detection can be done with "parity coding". In this method, a "parity bit" is inserted after each string of message bits of a predetermined length k so that the sum of the (k + l) bits is either always even (even parity) or always odd (odd parity). For example, if k = 4 and even parity is used, the message
110100101001. . . will become 110110010110010. . . On receiving the transmission signals, an error is detected if the sum of the appropriate five contiguous digits is odd. If the probability of error in a single bit is 1 percent, find (i) the probability of at least one error in the transmission of four sequential bits, and (ii) the probability of an undetected error after using even-parity coding.