STA1000S 2021
Week 1 Weekly Exercises – Set Theory & Basic Probability
Question 1:
1000 applicants have applied for a coding job at a data analytics company known as Data
Wizard. The applicants are proficient in different coding languages, but the company prefers
applicants who can code in 𝑅𝑅, 𝐩𝐩𝐩𝐩𝐩𝐩𝐩𝐩𝐩𝐩𝐩𝐩 or 𝑪𝑪++. We will use a three-variable Venn Diagram to
represent the problem. Let the top circle indicate the programmers who are proficient at
coding in 𝑅𝑅 (Denote this set with an R). Let the left circle indicate the programmers who are
proficient at coding in 𝐩𝐩𝐩𝐩𝐩𝐩𝐩𝐩𝐩𝐩𝐩𝐩 (Denote this set with a P). Finally, let the right circle indicate
the programmers who are proficient at coding in 𝑪𝑪++ (Denote this set with a C). The HR
department has determined the following information from the applicants:
• 40 applicants are proficient in all three coding languages.
• The total number of applicants that are proficient in 𝑅𝑅 = 595.
• The total number of applicants that are proficient in 𝐩𝐩𝐩𝐩𝐩𝐩𝐩𝐩𝐩𝐩𝐩𝐩 = 380.
• The total number of applicants that are proficient in 𝑪𝑪++ = 230.
• 89 applicants that applied were proficient in none of the three coding
languages.
• 121 applicants were proficient in both 𝑅𝑅 and 𝑪𝑪++.
• 74 applicants were proficient in 𝑅𝑅 and 𝐩𝐩𝐩𝐩𝐩𝐩𝐩𝐩𝐩𝐩𝐩𝐩 but not in C++.
a) Determine how many applicants are proficient at coding in 𝑅𝑅 but not in 𝐩𝐩𝐩𝐩𝐩𝐩𝐩𝐩𝐩𝐩𝐩𝐩 and
not in 𝑪𝑪++ (i.e., Determine 𝑛𝑛(𝑅𝑅 ∩ 𝑃𝑃� ∩ 𝐶𝐶̅ ) ).
b) Determine how many applicants are proficient at coding in 𝐩𝐩𝐩𝐩𝐩𝐩𝐩𝐩𝐩𝐩𝐩𝐩 and 𝑪𝑪++ but not
�) ).
in 𝑅𝑅 (i.e., Determine 𝑛𝑛(𝑃𝑃 ∩ C ∩ R
c) Suppose that applicants are randomly chosen. What is the probability that a
randomly chosen applicant would either be proficient at coding in 𝐩𝐩𝐩𝐩𝐩𝐩𝐩𝐩𝐩𝐩𝐩𝐩 but not in
𝑪𝑪++ and not in 𝑅𝑅 OR proficient at coding in 𝑪𝑪++ but not in 𝐩𝐩𝐩𝐩𝐩𝐩𝐩𝐩𝐩𝐩𝐩𝐩 and not in 𝑅𝑅
(Hint: First determine 𝑛𝑛(𝑃𝑃 ∩ 𝑅𝑅� ∩ 𝐶𝐶̅ ) and 𝑛𝑛(𝐶𝐶 ∩ 𝑅𝑅� ∩ 𝑃𝑃�) ).
Question 2:
Given that Pr(𝐴𝐴) = 0.45, Pr(𝐵𝐵) = 0.39 and Pr(𝐴𝐴 ∩ 𝐵𝐵) = 0.15, use the Venn Diagram below to
calculate the following probabilities:
a) Pr(𝐴𝐴̅ ∩ 𝐵𝐵� )
b) Pr�𝐵𝐵� ∩ (𝐴𝐴 ∪ 𝐵𝐵)�
(𝐴𝐴 ∪ 𝐵𝐵) ∪ ����������
c) Pr� ����������� (𝐴𝐴̅ ∪ 𝐵𝐵� ) �
, d) Pr� ����������
(𝐴𝐴̅ ∩ 𝐵𝐵� ) �
e) Pr(𝐴𝐴̅ ∪ 𝐵𝐵)
f) Pr(𝐴𝐴) + Pr(𝐵𝐵) − Pr(𝐴𝐴 ∪ 𝐵𝐵)
(Hint: Draw the Venn Diagram and shade in the specified regions for each question if you
are getting stuck.) Venn Diagram for above questions:
Question 3:
A domino is selected at random from a standard set of 28 dominoes (assume a blank is the
number 0 (zero)). The elements of the sample space, S, may be listed as:
Find the probability (3 decimal places) that:
Week 1 Weekly Exercises – Set Theory & Basic Probability
Question 1:
1000 applicants have applied for a coding job at a data analytics company known as Data
Wizard. The applicants are proficient in different coding languages, but the company prefers
applicants who can code in 𝑅𝑅, 𝐩𝐩𝐩𝐩𝐩𝐩𝐩𝐩𝐩𝐩𝐩𝐩 or 𝑪𝑪++. We will use a three-variable Venn Diagram to
represent the problem. Let the top circle indicate the programmers who are proficient at
coding in 𝑅𝑅 (Denote this set with an R). Let the left circle indicate the programmers who are
proficient at coding in 𝐩𝐩𝐩𝐩𝐩𝐩𝐩𝐩𝐩𝐩𝐩𝐩 (Denote this set with a P). Finally, let the right circle indicate
the programmers who are proficient at coding in 𝑪𝑪++ (Denote this set with a C). The HR
department has determined the following information from the applicants:
• 40 applicants are proficient in all three coding languages.
• The total number of applicants that are proficient in 𝑅𝑅 = 595.
• The total number of applicants that are proficient in 𝐩𝐩𝐩𝐩𝐩𝐩𝐩𝐩𝐩𝐩𝐩𝐩 = 380.
• The total number of applicants that are proficient in 𝑪𝑪++ = 230.
• 89 applicants that applied were proficient in none of the three coding
languages.
• 121 applicants were proficient in both 𝑅𝑅 and 𝑪𝑪++.
• 74 applicants were proficient in 𝑅𝑅 and 𝐩𝐩𝐩𝐩𝐩𝐩𝐩𝐩𝐩𝐩𝐩𝐩 but not in C++.
a) Determine how many applicants are proficient at coding in 𝑅𝑅 but not in 𝐩𝐩𝐩𝐩𝐩𝐩𝐩𝐩𝐩𝐩𝐩𝐩 and
not in 𝑪𝑪++ (i.e., Determine 𝑛𝑛(𝑅𝑅 ∩ 𝑃𝑃� ∩ 𝐶𝐶̅ ) ).
b) Determine how many applicants are proficient at coding in 𝐩𝐩𝐩𝐩𝐩𝐩𝐩𝐩𝐩𝐩𝐩𝐩 and 𝑪𝑪++ but not
�) ).
in 𝑅𝑅 (i.e., Determine 𝑛𝑛(𝑃𝑃 ∩ C ∩ R
c) Suppose that applicants are randomly chosen. What is the probability that a
randomly chosen applicant would either be proficient at coding in 𝐩𝐩𝐩𝐩𝐩𝐩𝐩𝐩𝐩𝐩𝐩𝐩 but not in
𝑪𝑪++ and not in 𝑅𝑅 OR proficient at coding in 𝑪𝑪++ but not in 𝐩𝐩𝐩𝐩𝐩𝐩𝐩𝐩𝐩𝐩𝐩𝐩 and not in 𝑅𝑅
(Hint: First determine 𝑛𝑛(𝑃𝑃 ∩ 𝑅𝑅� ∩ 𝐶𝐶̅ ) and 𝑛𝑛(𝐶𝐶 ∩ 𝑅𝑅� ∩ 𝑃𝑃�) ).
Question 2:
Given that Pr(𝐴𝐴) = 0.45, Pr(𝐵𝐵) = 0.39 and Pr(𝐴𝐴 ∩ 𝐵𝐵) = 0.15, use the Venn Diagram below to
calculate the following probabilities:
a) Pr(𝐴𝐴̅ ∩ 𝐵𝐵� )
b) Pr�𝐵𝐵� ∩ (𝐴𝐴 ∪ 𝐵𝐵)�
(𝐴𝐴 ∪ 𝐵𝐵) ∪ ����������
c) Pr� ����������� (𝐴𝐴̅ ∪ 𝐵𝐵� ) �
, d) Pr� ����������
(𝐴𝐴̅ ∩ 𝐵𝐵� ) �
e) Pr(𝐴𝐴̅ ∪ 𝐵𝐵)
f) Pr(𝐴𝐴) + Pr(𝐵𝐵) − Pr(𝐴𝐴 ∪ 𝐵𝐵)
(Hint: Draw the Venn Diagram and shade in the specified regions for each question if you
are getting stuck.) Venn Diagram for above questions:
Question 3:
A domino is selected at random from a standard set of 28 dominoes (assume a blank is the
number 0 (zero)). The elements of the sample space, S, may be listed as:
Find the probability (3 decimal places) that:
