Transcript 3.1 Set Notation

3.1
Set Notation
3.1.1 Venn Diagrams
Venn Diagram is used to illustrate
the idea of sets and subsets.
Example 1
XU
X
(b) A  B
U
U
B
A
• (c)there exists an element a such that a  X
and a  Y. or (X  Y and Y  X)
X
Y
3.1.2
Operations on
Sets
• Intersection
• Union
• Complement
Intersection
X  Y = {x : x  X and x  Y}
where  means “intersection”.
Example 1
Given X = {2,3,4,6,7,8, 10} and Y= {4,5,-2, 6, 9, 10}. Find X  Y.
X and Y are disjoint if X Y = .
– where  denotes the empty set.
– When X and Y are disjoint, the Venn
Diagram of X  Y is
X
Y
Union
X  Y = {x : x  X or x  Y}
• The Venn Diagram is
X
Y
U
• Example 1
Let A = {3,5,8,9,10} and E = {12,4, 3,
5, 10,24, 9}. Find A E and AE.
Complement
• We use or to denote the
complement of X.
= {x : x  U and x  X}

X
• In addition, we use Y\X to denote
the relative complement of X w.r.t. Y.
Y\X = {y : y  Y and y  Y}
• Example 1
• Please mark
in the following
diagrams to indicate the relative
complement of A w.r.t. B.
U
U
A
B
A
B
• Example 1
– Consider a deck of playing cards.
Let U be the set of all the cards.
R be the set of all the red cards.
D be the set of all the diamond
cards.
What , D\R and R\D? Illustrate these
sets with a Venn diagram.
3.2
•
Number of
Elements
For any two sets A and B , we have:
n(AB) = n(A) + n(B) – n(AB)
• Example 1
• Of the 70 S6 students of a school, 39 studied
Mathematics and Statistics(M), 37 studied
Geography (G), 42 studied History (H), 24 studied
both M and G, 26 studied both M and H, 25
studied both G and H, 18 studied all three
subjects.
• Find the number of students who study (a) at
least one of the three subjects, (b) none of the
three subjects.
3.3 Probability
•
Relative Frequency Definition of Probability
Suppose that a random experiment is repeated a large number
of times N, and that the event A occurs n times. Then the
probability of A is the limiting value of the relative frequency as
N becomes very large.
•
•
•
Weaknesses of the relative frequency definition
It requires a large number of repetitions of an experiment to
establish the probability of an even.
It assumes that the relative frequency will tend to a LIMIT.
•
Some Properties of Probability
For every event A in the sample sapce S,
1.
0  P(A)  1
2.
P(S) = 1
3.
If A and B are mutually exclusive events in S, then
P(AB) = P(A) + P(B)
***
P(impossible event) = 0
P(the certain event) = 1
• Law for Complementary Events
P(A’) = 1 – P(A)
Example
• A card is drawn at random from an ordinary pack of
52 playing cards. Find the probability that the card
(a) is a seven, (b) is not a seven.
3.4 Methods of
Counting
• The Multiplication Principle
– [Please refer to your F.6 Textbook]
• Permutations
– [Please refer to your F.6 Textbook]
• Combinations
– [Please refer to your F.6 Textbook
Combinations
•
Example 1
•
Example 2
•
Example 3
•
Example 4
– If the letters of the word “MINIMUM” are arranged in a line at
random, what is the probability that the three M’s are together at the
beginning of the arrangement?
– Ten pupils are placed at random in a line. What is the probability that
the two youngest pupils are separated?
– If a four-digit number is formed form the digits 1,2,3 and 5 and
repetitions are NOT allowed, find the probability that the number is
divisible by 5?
– In how many ways can a hand of 4 cards be dealt from an ordinary pack
of 52 palying cards?
•
Example 5
•
Example 6
•
Example 7
– Four letters are chose at random from the word RANDOMLY. Find the
probability that all four letters chosen are consonants.
– A team of 4 is chosen at random from 5 girls and 6 boys.
– In how many ways can the team be chosen if
(i) there is are no restrictions;
(ii) there must be more boys than girls?
– Find the probability that the team contains only one boy.
– Four items are taken at random from a box of 12 items and inspected.
The box is rejected if more than 1item is found to be faulty. If there
are 3 faulty items in the box, find the probability that the box is
accepted.