Description
problem 1. We roll two standard 6-sided dice. In each part, determine whether the two events are
independent.
(a) Let A = ”the sum of of the two rolls is odd” and B = ”both tosses were greater than 3.”
(b) Let C = ”the two rolls showed the same number” and D = ”the second toss was greater than 4.”
problem 2. A student applies to both BU and Northeastern. Looking at statistics from students who
applied to the same schools in the past, she figures out that she has a probability of 0.25 of getting into BU
and 0.5 of getting into Northeastern. She also finds that she has a probability of 0.2 of getting into both.
(a) What is the probability that she gets into Northeastern given that she gets admitted to BU?
(b) Are the two events (“getting into BU” and “getting into Northeastern”) independent?
(c) How do you think your answer to the previous question might reflect reality?
problem 3. Let A and B be two independent events. Prove that:
(a) A and B are independent.
(b) A and B are independent.
Let Pr(A) = p and Pr(B) = q. Find each of the following probabilities, expressed in terms of p and q:
(c) Pr(A | B).
(d) Pr(A ∪ B).
problem 4. 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 1
4
, and it is correctly transmitted with probability 3
4
; a 1 bit is incorrectly
transmitted as a 0 with probability 1
3
, and it is correctly transmitted with probability 2
3
; each bit is flipped
independently from the other bits.
(a) Alice chooses a single bit uniformly at random and sends it to Bob. What is the probability that Bob
receives it correctly?
(b) What is the probability that Bob receives the message 1011 correctly?
(c) 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?
(d) Alice chooses a single bit uniformly at random and she uses the scheme in part (c) to send it. What
is the probability that the bit was 0 given that Bob received the sequence 101?
problem 5. Tang Sanzang is a Chinese monk travelling to the west to seek Buddhist scriptures, accompanied by his monster companions Wukong, Bajie, and Wujing. The fellowship encounters a crossroads with
three paths (starting at A, B, and C) which cut through a dangerous mountain range filled with hungry
monsters, however, they all lead to the same exit Z. They happen to run into a travelling oracle, who
draws them a map of the paths, and supplies the probability of survival for various segments. However,
he is unable at that moment to measure of the probability of every possible segment. The map produced
by the oracle is as shown below:
A D
B E Z
C
4/5
25/28
3/4
7/10
49/100
In the above map, the solid lines are the segments on which the fellowship can travel on (e.g., the path
B Z from B to Z is comprised of the segments B → D → E → Z). The dashed lines provide the
probability of survival on the entire path between the endpoints (e.g., the probability of survival on the
path B → D → E is 3/4). For any segment, the probability of surviving on that segment is independent
of other segments (recall that the segments are the solid lines).
For each of the three paths A Z, B Z, C Z, find the probability of survival on that path. Which
path should the fellowship take in the interest of survival?
H7-2