本次加拿大代写是一个概率学的限时测试

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

received correctly.

(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

received correctly.

(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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