Alice wants to send a message to Bob, where the message is a sequence of bits. A

Question

Alice wants to send a message to Bob, where the message is a sequence of bits. Alice sends her message through a noisy communication channel that randomly flips the bits: a 0 bit is incorrectly transmitted as a 1 with probability eo, and it is correctly transmitted with probability 1- E0; a 1 bit is incorrectly transmitted as a 0 with probability e1, and it is correctly transmitted with probability 1 - e1; each bit is flipped independently from the other bit:s 1. Alice chooses a single bit uniformly at random and sends it to Bob. What is the probability that Bob receives it correctly? 2. What is the probability that Bob receives the message 1011 correctly? 3. In an effort to improve the probability that Bob receives the correct message, Alice transmits each bit three times and Bob uses the majority rule to decode. More precisely, Alice transmits a 0 as 000 and a 1 as 111. Bob decodes the three bits received as a 0 if there are at least 2 0s, and as a 1 otherwise What is the probability that Bob correctly decodes a 0? 4. For what values of eo is there an improvement in the probability that Bob correctly decodes a 0 when Alice uses the scheme in part (3)? 5. Alice chooses a single bit uniformly at random and she uses the scheme in part (3) to send it. What is the probability that the bit was 0 given that Bob received the sequence 101?

Explanation / Answer

a) Probability that Bob received it correctly = 1-e

b)Probability that 1011 is received correctly =[1-e]^4

c)Probability that message is decoded correctly = 3C2 * [1-e]^2*e +[1-e]^3=[1-e]^2 *[1+2e]

d)Improvement is there is [1-e]^2 *[1+2e]>[1-e]

[1+2e]*[1-e]>1

1+e-2e^2>1

e=1/2

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