due fri feb 23, 2018 at 11am via gradescope H5.
The assignment is due at Gradescope on February 23 at 11am. Late assignments will not be accepted.
Submit early and often.
You are permitted to study with other students in the class and discuss the problems; however, you
must write up your own solutions, in your own words. Do not submit anything you cannot explain. If you
do collaborate with any of the other students on any problem, please do list all your collaborators in
your submission for each problem.
Finding solutions to homework problems on the web, or by asking students not enrolled in the class
is strictly prohibited.
Unless otherwise specified, you can leave your answer in closed form. (e.g. 1 (1207 )(0.1)200).
Show your work. Answers without justification will be given little credit. Justify each step in your
solutions e.g. by stating that the step follows from an axiom of probability, a definition, algebra, etc.;
for example, your answer could include a line like this:
Pr(X\Y\Z) Pr(A[B) = Pr(X\Y\Z) (Pr(A) + Pr(B)) (A and B are disjoint)
The syllabus has some pointers on using LaTeX and Python.
problem 1 (Tricky coins)
Alice has 5 coins. Two of the coins are double-headed (both sides are heads). One of the coins is
double-tailed (both sides are tails). The remaining two coins are normal (one side is heads and one side is
tails). The normal coins are fair.
Alice picks one of the coins uniformly at random, closes her eyes, and she tosses the coin. What is the
probability that the lower face of the coin is a heads?
Alice opens her eyes and sees that the upper face of the coin is heads. What is the probability that the
lower face of the coin is heads?
Alice closes her eyes again, and tosses the same coin a second time. What is the probability that the
lower face is heads?
Alice opens her eyes and sees that the upper face of the coin is heads. What is the probability that the
lower face is heads?
Alice discards this coin. She picks a coin uniformly at random from the remaining coins and tosses it.
What is the probability that the coin shows heads?
Hint: all of the questions except the first one ask you to work out a conditional probability. Carefully
identify the event on which we are conditioning. What has Alice learned from the previous coin tosses?
Solution: Your solution goes here.
H5-2
problem 2 (Independent coins)
We toss 3 fair and mutually independent coins. Consider the following events:
Let A be the event that the first coin is heads.
Let B be the event that the second coin is heads.
Let C be the event that the third coin is heads.
Let D be the event that an even number of coins are heads.
1. Find the sample space for this experiment and the probability of each of the events A, B, C, D.
2. Show that the events A, B, C, D are not mutually independent.
3. Show that any subset of three events from the setfA, B, C, Dgare mutually independent.
Solution: Your solution goes here.
H5-3
problem 3 (Sending bits through a noisy channel)
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 e0, 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 bits.
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 e0 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?
Solution: Your solution goes here.