PROBABILITY I

Sample Space

Important concept:

·         An experiment is a process or action to observe its outcomes.

·         A sample space is the set of all possible outcomes from an experiment. It can be represented in the form of set notation.

·         Example : S ={(1,2),(1,3),(1,4)}

Example 1

Experiment: Tossing a fair coin

Sample space,

 S = {Head, Tail}

Example 2

Experiment:  A fair six-sided die is rolled.

Sample space, S ={1,2,3,4,5,6}

Example 3

Experiment: Drawing a ball from a bag containing three balls of the same size: white, black and green.

Sample space, S = {white ball, black ball, green ball}

STATISTICS I AND II

            

   MODE

The mode is the most common value or the item with the highest frequency

Examples

Find the mode for the following set of data.

a.         2,  1,  3,  2,  5,  3,  2,  5,  6,  3,  3,  4

 

   

            Mode = 3

b.         7,  4,  3,  4,  8,  7,  5,  4,  3

            Mode = 4

c.        

Number Of goals	0	1	2	3	4
Frequency	3	6	7	5	2

 

            Mode = 2

d.        

Grade	A	B	C	D	E
Frequency	6	8	7	6	2

            Mode = Grade B

Exercice  8.2

a.         20 kg,  50 kg,  30 kg,  70 kg, 50 kg, 

            40 kg,  30 kg,  60 kg,  50 kg,  60 kg,

            50 kg,  60 kg

            Mode  =  _____ kg

b.         15 cm,  18 cm,  12 cm,  13 cm,  15 cm, 

17 cm,  12 cm,  16 cm,  15 cm,  13 cm, 

14 cm,  15 cm

Mode  = ______  cm

           

c.         4,  5,  4,  2,  1,  4,  6,  7,  5,  6

            Mode  =

d.        

Number of matches played	1	2	3	4	5
Frequency	1	2	5	3	3

            Mode  =  _______matches

e.        

Height ( cm )	120	121	122	123	124
Frequency	5	6	8	12	9

           

Mode  =         cm

f.         

Number Of goals	0	1	2	3	4
Frequency	2	5	4	4	3

            Mode  =          goals

8.3       MEDIAN

The median is the value which is located in the middle of the set of values that has been arranged in increasing or decreasing order.

Examples

Determine the median from the following sets of data.

a.         2, 6, 9, 4, 3, 4, 5, 6, 7

 

            2,  3,  4,  4,  5,  6,  6,  7,  9

            Median  =  5

b.         9,  7,  3,  4,  8,  6,  7

            =  3,  4,  6,  7,  7,  8,  9

            Median  =  7   

c..        10kg,  12kg,  18kg,  10kg,  16kg

            =  10,  10,  12,  16,  18

            Median  =  12kg

d.         RM5,  RM7,  RM9,  RM6,  RM8,  RM6,     

             RM10,  RM9

            =  5,    6,    6,    7,    8,    9,    9,    10

            Median  =

                        =  RM7.50

Exercise 8.3

a.         3,  2,  3,  4,  7,  2,  1

            Median  =

b.         9,  2,  5,  1,  8,  9,  0

            Median  =

c.         2kg,  4kg,  1kg,  5kg,  4kg,  3kg

            Median  =

d.         20cm,  10cm,  30cm,  10cm,  40cm

            Median  =

e.          

Number Of goals	0	1	2	3	4
Frequency	2	5	4	4	2

            Total frequency  = 2 + 5 + 4 + 4 + 2  =

Median  =  Number of goals of the  ___ th

                  Frequency

             =       goals

 

 f.        

Grade	A	B	C	D	E
Numbers of students	6	12	9	8	2

            Total frequency  = 6 + 12 + 9 + 8 + 2  =

Median  =  Grade of the ____th students

             =      

8.4       MEAN

Mean is the sum of all the values divided by the number of data  

                   
When a set of data is given in a frequency table,

ALGEBRAIC EXPRESSION I, II, III

·         Unknown

·         Important Concept

·         ‘An unknown is a quantity whose value has not been determined’

            4.1.1

Example		Exercise
Azmi bought some durians       Solution:             The unknown is the number of   durian.	1. 2. 3. 4. 5.	There were a lot of people in the fun fair. Solution: Mr.Tan lost a large amount of money when he dropped his wallet Solution: A number of students are studying in the library. Solution: My father bought a few kilograms of rambutans yesterday. Solution: Ketty gained a few kilograms of weight during the holidays Solution:

• Algebraic term

·         identifying algebraic terms with one unknown.

·         Important  concept:

·         ‘An algebraic term with one unknown is the product of a number and an unknown’  

 4.1.2

	EXAMPLE		EXERCISE
	is the multiplication of the number   6 with the unknown	1. 2.	Determine the algebraic term with one unknown for each of the following. 3e Solution: -5t Solution:

LINEAR EQUATIONS

3.1 Solutions of Linear Equations in One Unknown

·         Solve linear equations = Finding the value of the unknown which satisfies the equation.

·         The solution of the equation is also known as the root of the equation.

·         A linear equation in one unknown has only one root.

·         To determine whether a given value is a solution of an equation, substitute the value into the equation. If the sum of the left hand side (LHS) = sum of right hand side (RHS), then the given value is a solution.

3.2 Solving Linear Equations in One Unknown through the operations  +,  –   ,   and

            There are 4 different forms of linear equation as follows :

Type	Equation	Operation	Solution
I	x + a = b	–	x = b – a
II	x – a = b	+	x = b + a
III	ax = b		x =
IV			x = a b

TRANSFORMATION I, II

2.1 Translation       1.   Translation is a transformation that moves all the points on a plane through the same               distance and in the same direction.            2.   Properties of a translation             a)   the shape, size and orientation of the object and the image are the same             b)   every point is moved through the same distance and in the same direction       3.   A translation is usually  expressed in the form , where h  represents the horizontal             movement parallel to the x-axis and k represents the vertical movement parallel to             the y- axis

GRAPHS OF FUNCTION II

Important Notes:

For Paper 1 (Objective Questions), the questions asked are usually related to the  shapes and positions of basic graphs of LINEAR, QUADRATIC, CUBIC and RECIPROCAL FUNCTIONS.

Students are expected to be able to identify:

        a)    the shape of graph given a type of function

        b)    the type of function given a graph

        c)    the graph given a function and vice versa.

The different types of functions for Paper 1, SPM Mathematics:

Linear y = mx + c m = gradient c = y-intercept	Quadratic y = ax2 + bx + c Note : Paper 1 limits to            y =  ax2 + c	Cubic y = ax3 +bx2 +cx + d Note : Paper 1 limits to        y = ax3 + c	Reciprocal y =1 x
Note :  This module concentrates on graph sketching, the skills needed for Paper 1.  For details on how to plot graphs please refer to the other related modules (eg. Chapter 12 – The Straight Line – on how to draw /  plot straight lines)

19.0 Back to Basic

I.  Linear Graphs  :  can be represented by the equation  y = mx + c.

·         m is the gradient

·         c the y-intercept, ie, the place where the straight line cuts the y-axis.

II. The general form of a Quadratic Function is f(x) = ax² + bx + c;  a, b, c are constants and  a ≠ 0.

     Characteristics of a quadratic function:

·         Involves one variable only,

·         The highest power of the variable is 2.

III. The general form of a Cubic Function is f(x) = ax³ + bx² + cx + d ;  a, b, c and d are constants and  a ≠ 0.

     Characteristics of a cubic function:

·         Involves one variable only,

·         The highest power of the variable is 3.

IV.  The simple Reciprocal Function is of the form   y =  ,  where a is a constant.

19.1  To Identify The Type Of Function Given Its Equation

Example 1

No.	Linear y = mx + c	Quadratic y = ax2 + bx + c	Cubic y = ax3 + bx2 + cx + d	Reciprocal y =
1.	y  = x	y = x2 + 2x -3	y = x3 +  2x2 + 3x – 4	y =
2.	y  =  2x + 3	y = x2 + 2	y = x3 – 3	y =
3.	y  =  3x – 4	y =  4 – 3x2	y =  4 – 3x3	xy  =  4
4.	y  =  5	y = 3x2	y = 2x3	y =
5.	x+ y  =  4	y  =  3 – 4x + 5x2	y  =  1 – 2x + 5x3	y =