Name: _______________________
Class: __________
Date:_____________
Math SL: 15I Sets and Venn diagrams
Review:
4
3ö
æ
1. a. Expand ç x - ÷ and simplify each term.
è
xø
æ
b. Find the constant term when (1+ x ) ç x è
4
3ö
÷ is expanded.
xø
2. A company produces cell phones that use components A, B and C. The probabilities that each
of these components are faulty are 0.02, 0.1 and 0.08 respectively.
a. Find the probability that none of the components in a cell phone is faulty.
b. Find the probability that at least one component in a computer is faulty.
Ten cell phones are assembled. If any of the components is faulty, then the cell phone does not
work.
c. Find the probability that exactly two of the cell phones do not work.
d. Find the probability that at most two of the cell phones do not work.
1
15I Sets and Venn Diagrams
Today’s Objective:
(1) to introduce Venn diagrams as a method to solve certain kinds of probability problems
Venn Diagrams
A Venn diagram is a useful way of comparing two or more events in a sample space.
The ____________________ set, or sample space U, is represented by a ___________________. An event A is
represented by a _____________.
Notation
x ÎA means “x is an ________________ of the set A.”
n(A) means “the ________________ of elements in set A.”
Æ is the symbol for the _______________ set.
Set Notation
A’
Meaning
Venn Diagram
All elements not in A.
the complement
of set A
AÈB
the union of sets
A and B
AÇB
the intersection
of sets A and B
All elements belonging to A or B
or both A and B.
In set notation
A È B = { x | x ÎA or x ÎB}
All elements common to sets A
and B. Must be in A and B.
In set notation
A Ç B = { x | x ÎA and x ÎB}
AÇB = Æ
Sets which do not have elements
in common.
disjoint sets
Also called mutually exclusive.
Note: You should be able to create any Venn diagram using the new math symbols above.
e.g. ( A Ç B ) ', A'Ç B' , A'È B' , etc.
2
Example 1: Let A be the set of all factors of 6, and B be the set of all positive even integers < 11.
a) Describe A and B using set notation.
b) Find:
i. n(A)
ii. A È B
iii. A Ç B
Example 2: In a class of 40 students, 19 play tennis, 20 play netball, and 8 play neither of these
sports. A student is randomly chosen from the class. Determine the probability that the student:
a) plays tennis
b) does not play netball
c) plays at least one of the sports
d) plays one and only one of the sports
e) plays netball but not tennis
f) plays tennis given he or she plays netball.
Example 3: Use Venn diagrams to verify that ( A Ç B)' = A'È B'.
3
Example 4: Human blood is classified O, A, B, or AB, depending upon whether the blood contains
no antigen, an A antigen, a B antigen, or both the A and B antigens. A third antigen, called Rh
antigen, is important to human reproduction. Blood is said to be Rh-positive if it contains this
Rh-antigen; otherwise, the blood is Rh-negative. So for instance, blood is type A+ if it contains
the A antigen and the Rh antigen. Blood is type AB- if it contains the A and B antigens but does
not contain the Rh antigen.
 22 people were either type A or type AB, 16 of which had the Rh antigen
 27 were either type B or type AB, 18 of which had the Rh antigen
 8 were type AB, 2 of which were AB 35 were type O, 5 of which were Oa.
b.
c.
d.
How many patients are listed here?
How many patients have type B+ blood?
How many patients have type A- blood?
How many have exactly two antigens?
Hmwk#46
15I Sets and Venn diagrams
pg. 449 # 2, 3, 4, 6, 7, 8
pg. 451 # 1(c), 3
10H
pg. 263 # 4
Continue researching a math exploration topic – find math
involved.
4