When a message is sent electronically it is usually sent as astream of bits, each of which can be either a 0 or a 1. If thedigital channel is noisy then each
bit has some probability of being flipped (ie changed from a 0 to a1 or vice versa) resulting in a corrupted message.
Assume that a message is being sent through a noisy channel wherethe probability that any individual bit will be flipped is 0.1.What is the probability that
a message 4 bits long would be successfully transmitted? (Answer tothree decimal places).
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One method of dealing with the problem of bits being flipped is touse a Hamming code. This involves sending extra bits along with themessage that can
be used to check the main message. For example a 7 bit Hamming Codecontains 4 bits of message data and 3 check bits. If only one ofthe bits is in
error at the receiving end then mathematical techniques can be usedto determine which one it is and apply a correction. However, ifmore than one bit is
flipped then an erroneous correction will be applied and themessage will still be corrupted.
Assume that a message is being sent through a noisy channel wherethe probability that any individual bit will be flipped is 0.1 asbefore. If the message is
sent using a 7 bit hamming code what is the probability that itwill get through with no more then one of the seven bits beingflipped? (Answer to three
decimal places).
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If we are concerned with the possibility that 2 bits have beenflipped then instead of using the correction mentioned above we canask for the 7 bits to be
re-transmitted. However, there is the possibility that when morethen 3 bits have been flipped that the final corrupted message willcorrespond to a
message where the Hamming code thinks that none of the bits havebeen flipped. The probability of this occurring when 3 or 4 bitshave been flipped is
0.2. It does not occur if 5 or 6 bits have been flipped (similar to1 or 2 bits being flipped). It always occurs if all 7 of the bitshave been flipped.
Assume that a message is being sent through a noisy channel wherethe probability that any individual bit will be flipped is 0.1 asbefore. A 7 bit Hamming
code is being used. When the message is decoded at the receivingend the procedure indicates that none of the bits have beenflipped. What is the
probability that the message that was received that appears to haveno bits flipped is in fact an error with 3 or more bits flipped?(Answer to three decimal
places).
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Assume that a message is being sent through a noisy channel wherethe probability that any individual bit will be flipped is 0.1 asbefore. A 7 bit Hamming
code is being used. When the message is decoded at the receivingend the procedure indicates that at least one of the bits have beenflipped. Therefore
the 7 bit message is re-transmitted. What is the probability thatif the decoding procedure is applied to both transmissions thatboth times will result in the
correct 7 bit message (including the case where the secondtransmission does not need any unflipping)? (Answer to threedecimal places).
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