In many cases in networks, instead of using all 8 bits for data, we often utilize only 7 bits for data and one bit for 'control'.
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In many cases in networks, instead of using all 8 bits for data, we often utilize only

7 bits for data and

one bit for 'control'. Specifically, this control bit is an even-parity bit to

ensure that the number of bit '1' in total 8 bits-long data is always even.

Assume a binary symmetric channel with bit error probability pe as in the lecture note. All bit

errors are independent of another.

Sender A now transmits an 8-bit long packet (7 data bits + one even-parity bit) onto this binary

symmetric channel. The receiver B receives a packet and there are total three possibilities:

• Case 1: no bit has been inverted (no error) so the receiver successfully receives the packet.

• Case 2: Some bits are inverted (error!) and the receiver can correctly 'detect' the error.

• Case 3: Some bits are inverted (error!) but the receiver mistakenly concludes that the

packet is (or appears to be) error-free.

Find the probability of each of these three cases (5 points each). No need to find a closed form.

Just leave your expression as is. Note that the sum of the probabilities of these 3 cases should

be equal to one.

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