3.1Set Notation Venn Diagrams Venn Diagram is used to illustrate the idea of sets and subsets. Example 1 X  U(b) A  B X U B A U

(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 where  means “ intersection ”. X  Y = {x : x  X and x  Y} Example 1 Given X = {2,3,4,6,7,8, 10} and Y= {4,5,-2, 6, 9, 10}. Find X  Y.

–where  denotes the empty set. –When X and Y are disjoint, the Venn Diagram of X  Y is X and Y are disjoint if X  Y = . X Y

Union The Venn Diagram is X  Y = {x : x  X or x  Y} 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. In addition, we use Y\X to denote the relative complement of X w.r.t. Y. = {x : x  U and x  X} 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. A B U U 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.

3.2Number 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 –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? Example 2 –Ten pupils are placed at random in a line. What is the probability that the two youngest pupils are separated? Example 3 –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? Example 4 –In how many ways can a hand of 4 cards be dealt from an ordinary pack of 52 palying cards?

Example 5 –Four letters are chose at random from the word RANDOMLY. Find the probability that all four letters chosen are consonants. Example 6 –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. Example 7 –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.