We have already discussed parity check code in the class. The problem is that it cannot detect even-bit errors. In order to handle this issue, we design an advanced parity check as follows: We use multiple parity bits and each parity bit is generated from a different subset of the original data bits. For example, consider the original data have 6 bits. We add two parity bits: The first one is the parity bit of 1st, 2nd, 3rd, and 4th bits; the second one is the parity bit of 1st, 2nd, 5th, and 6th bits. If the original data is 111111, then, the parity bits will be 00. This code can also be represented in a table below. "Y” in column i and row j indicates the ith bit is in the subset to compute the ith parity. (In reality, this is called linear block code.)

 

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1 Parity 1 Parity 2 Y Y Y Y N Y Y Y Still we send the original data bits and parity bits to the receiver. Each bit is independently flipped with probability p. We define • Event A: None of the bits are flipped; • Event B: Some of the bits are flipped, and this is detected by the advanced parity check. • Event C: Some of the bits are flipped, but this is not detected by the advanced parity check. (1) Let p= 0.1. Through theoretical analysis, compute the probability of events A, B, and C, i.e., pA, PB, and pc. 1 Parity 1 Parity 2 Y Y Y Y N Y Y Y Still we send the original data bits and parity bits to the receiver. Each bit is independently flipped with probability p. We define • Event A: None of the bits are flipped; • Event B: Some of the bits are flipped, and this is detected by the advanced parity check. • Event C: Some of the bits are flipped, but this is not detected by the advanced parity check. (1) Let p= 0.1. Through theoretical analysis, compute the probability of events A, B, and C, i.e., pA, PB, and pc.