1. Consider a binary communication channel for exchanging messages encoded in bits, i.e. se
quences of 0’s and 1’s. The channel is \noisy”, in the sense that 0’s have a 10% chance of
being flipped to 1’s during transmission, and 1’s have a 5% chance of being flipped to 0’s,
independently of other bits in the message.

(a) (4 points) If you send the message (1; 0; 1) through the channel, find the probability it is

(b) (8 points) If you receive the message (1; 1), find the probability that this was the actual
message that was sent through the channel. For you answer, assume that all four 2-bit
messages, namely (0, 0), (0, 1), (1, 0), (1, 1), are equally likely to have been sent through
the channel.

2. Consider a binary communication channel for exchanging messages encoded in bits, i.e. se
quences of 0’s and 1’s. The channel is \noisy”, in the sense that 0’s have a 5% chance of
being flipped to 1’s during transmission, and 1’s have a 10% chance of being flipped to 0’s,
independently of other bits in the message.

(a) (4 points) If you send the message (1; 0; 1) through the channel, find the probability it is

(b) (8 points) If you receive the message (0; 1), find the probability that this was the actual
message that was sent through the channel. For you answer, assume that all four 2-bit
messages, namely (0, 0), (0, 1), (1, 0), (1, 1), are equally likely to have been sent through
the channel.

3. An exam consists of 4 questions, where each question has one of two possible variations.

(a) (2 points) How many distinct exams are there? (exams are distinct if at least one question
has different variations.)

(b) (4 points) Each student is randomly assigned one of the above versions of the exam. Find
the probability that two specific students get identical exams (i.e. all 4 questions have the
same variations).

(c) (6 points) Find the probability that two specific students get exams which differ by exactly
two questions (i.e. have 2 questions with the same variations, and 2 questions with different
variations).

4. An exam consists of 4 questions, where each question has one of two possible variations.

(a) (2 points) How many distinct exams are there? (exams are distinct if at least one question
has different variations.)

(b) (4 points) Each student is randomly assigned one of the above versions of the exam. Find
the probability that two specific students get at least one question in common (i.e. one or
more of the 4 questions have the same variation).

(c) (6 points) Find the probability that two specific students get exams which differ by exactly
one question (i.e. have 1 questions with the same variation, and 3 questions with different
variations).

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