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1-Problem
Suppose
that
the
universal
set SS is
defined
as S={1,2,⋯,10}S={1,2,⋯,10} and A={1,2,3}A={1,2,3},B={X∈ S:2≤X≤7}B
={X∈S:2≤X≤7}, and C={7,8,9,10}C={7,8,9,10}.
a.
b.
c.
d.
Find A∪ BA∪B.
Find (A∪C)−B(A∪C)−B.
Find A¯∪(B−C)A¯∪(B−C).
Do A,B,A,B, and CC form a partition of
SS?
2-Problem
When working with real numbers, our universal set is
following sets.
a.
b.
c.
d.
ℝR.
Find each of the
[6,8]∪[2,7)[6,8]∪[2,7)
[6,8]∩[2,7)[6,8]∩[2,7)
[0,1]c[0,1]c
[6,8]−(2,7)[6,8]−(2,7)
3-Problem
For each of the following Venn diagrams, write the set denoted by the shaded
area.
a.
b.
c.
d.
4-Problem
A coin is tossed twice. Let SS be the set of all possible pairs that can be
observed,
i.e.,S={H,T}×{H,T}={(H,H),(H,T),(T,H),(T,T)}.S={H,T}×{H,T}={(H,H),(H
,T),(T,H),(T,T)}. Write the following sets by listing their elements.
a.
b.
c.
AA: The first coin toss results in head.
BB: At least one tail is observed.
CC: The two coin tosses result in different outcomes.
5-Problem *
Let A={1,2,⋯,100}A={1,2,⋯,100}. For any i∈ ℕi∈N, Define
numbers in AA that are divisible by ii. For example:
A2={2,4,6,⋯,100},A2={2,4,6,⋯,100},
A3={3,6,9,⋯,99}.A3={3,6,9,⋯,99}.
a. Find
b. Find
AiAi as the set of
|A2||A2|,|A3||A3|,|A4||A4|,|A5||A5|.
|A2∪A3∪ A5||A2∪A3∪A5|.
6-Problem
Suppose that A1A1, A2A2, A3A3 form a partition of the universal set
Let BB be an arbitrary set. Assume that we know
|B∩A1|=10,|B∩A1|=10,
|B∩A2|=20,|B∩A2|=20,
|B∩A3|=15.|B∩A3|=15.
Find |B||B|.
SS.
7- Problem *
Let
AA and BB be two events such that
P(A)=0.4,P(B)=0.7,P(A∪B)=0.9P(A)=0.4,P(B)=0.7,P(A∪B)=0.9
a.
b.
c.
d.
e.
f.
Find
Find
Find
Find
Find
Find
P(A∩B)P(A∩B).
P(Ac∩B)P(Ac∩B).
P(A−B)P(A−B).
P(Ac−B)P(Ac−B).
P(Ac∪B)P(Ac∪B).
P(A∩(B∪Ac))P(A∩(B∪Ac)).
8- Problem
Suppose that, of all the customers at a coffee shop,
•
•
•
70%70% purchase a cup of coffee;
40%40% purchase a piece of cake;
20%20% purchase both a cup of coffee and a piece of cake.
Given that a randomly chosen customer has purchased a piece of cake, what
is the probability that he/she has also purchased a cup of coffee?
9- Problem
Let
A,BA,B,
a.
b.
c.
d.
Find
Find
Find
Find
and
CC be
three
events
with
probabilities
given
P(A|B)P(A|B).
P(C|B)P(C|B).
P(B|A∪C)P(B|A∪C).
P(B|A,C)=P(B|A∩C)P(B|A,C)=P(B|A∩C).
10-Problem
You choose a point
(X,Y)(X,Y) uniformly at random in the unit square
S={(x,y)∈ ℝ2:0≤x≤1,0≤y≤1}.S={(x,y)∈R2:0≤x≤1,0≤y≤1}.
Let AA be the event {(x,y)∈ S:|x−y|≤12}{(x,y)∈S:|x−y|≤12} and BB be the
event {(x,y)∈ S:y≥x}{(x,y)∈S:y≥x}.
a. Show sets AA and BB in the x-y plane.
b. Find P(A)P(A) and P(B)P(B).
c. Are AA and BB independent?
11-Problem
below:
One way to design a spam filter is to look at the words in an email. In
particular, some words are more frequent in spam emails. Suppose that we
have the following information:
•
•
•
50%50% of emails are spam;
1%1% of spam emails contain the word “refinance”;
0.001%0.001% of non-spam emails contain the word “refinance”.
Suppose that an email is checked and found to contain the word “refinance”.
What is the probability that the email is spam?
12-Problem
I toss a fair die twice, and obtain two numbers XX and YY. Let AA be the
event that X=2X=2, BB be the event that X+Y=7X+Y=7, and CC be the
event that Y=3Y=3.
a.
b.
c.
d.
Are
Are
Are
Are
AA and BB independent?
AA and CC independent?
BB and CC independent?
AA, BB, and CC are independent?
13-Problem
A coffee shop has 44 different types of coffee. You can order your coffee in a
small, medium, or large cup. You can also choose whether you want to add
cream, sugar, or milk (any combination is possible, for example, you can
choose to add all three). In how many ways can you order your coffee?
14-Problem
Eight committee members are meeting in a room that has twelve chairs. In
how many ways can they sit in the chairs?
15-Problem
There are 2020 black cell phones and 3030 white cell phones in a store. An
employee takes 1010 phones at random. Find the probability that
a. there will be exactly 44 black cell phones among the chosen phones;
b. there will be less than 33 black cell phones among the chosen phones.
16-Problem
Five cards are dealt from a shuffled deck. What is the probability that the dealt
hand contains
a. exactly one ace;
b. at least one ace?
17-Problem
Five cards are dealt from a shuffled deck. What is the probability that the dealt
hand contains exactly two aces, given that we know it contains at least one
ace?
18-Problem
There are 5050 students in a class and the professor chooses 1515 students
at random. What is the probability that you or your friend Joe are among the
chosen students?
19-Problem
You
coin
have a biased coin for which P(H)=pP(H)=p.
2020 times. What is the probability that
a. you observe 88 heads and
1212 tails;
You
toss
the
20-Problem
b. You roll a die 55 times. What is the probability that at least one value is
observed more than once?
…
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