Math Revision Dse

Revision Summary of DSE Mathematics (Compulsory Part)

Number and Algebra

Page 1

Polynomials

Like terms and unlike terms

e.g. 2x and 6x are like terms and 2x + 6x can be simplified as (2 + 6)x = 8x 4pq2 and pq2 are like terms and 4pq2 pq2 can be simplified as (4 1)pq2 = 3pq2 6ab and 5ac are unlike terms and they cannot be simplified

Factorization

1. 2 2 ( )( )a b a b a b + 2. 2 2 22 ( )a ab b a b+ + + 3. 2 2 22 ( )a ab b a b + 4. 3 3 2 2( )( )a b a b a ab b+ + + 5. 3 3 2 2( )( )a b a b a ab b + +

Remainder and Factor Theorems

Remainder theorem When a polynomial ( )f x is divided by a linear polynomial ax b+ where 0a , the remainder

is ( )bfa

.

Factor theorem

If ( )f x is a polynomial and ( ) 0bfa

= , then the polynomial ax b+ is a factor of the

polynomial ( )f x . Conversely, if a polynomial ax b+ is a factor of polynomial ( )f x , then ( ) 0bf

a

= .

Let 3 2( ) 4 3 18f x x x x= + . (a) Find (2)f .

3 2(2) (2) 4(2) 3(2) 18 0f = + = (b) Factorize ( )f x . Since (2) 0f = , 2x is a factor of ( )f x .

3 2 2 24 3 18 ( 2)( 6 9) ( 2)( 3)x x x x x x x x+ = + + = +

Let ( ) ( 3)(2 1) 3f x x x x= + + + . Find the remainder when ( )f x is divided by 3x + . The required remainder [ ][ ]( 3) ( 3) 3 2( 3) 1 ( 3) 3 6( 5) 30f= = + + + = =

Number and Algebra

Page 2

Solving Quadratic Equations in One Unknown

Factor method e.g. Solve x2 + 2x 3 = 0.

1 or 30 1or 03

0)1)(3(0322

==

==+

=+

=+

x x

xx

xx

xx

Completing the square

Convert it to the form of 2 2( 2 )a x px p q+ + = , 2( )a x p q+ = , qx pa

=

Quadratic formula e.g. Solve x2 7x + 9 = 0.

Using the quadratic formula, 2 2( 7) ( 7) 4(1)(9) 4

Substitute 1, 7 and 9 into .2(1) 2

7 13 7 13 7 13 i.e. or

2 2 2

b b acx a b c x

a

= = = = =

+ =

0 a < 0

The graph opens upwards.

The graph opens downwards. Finding the Optimum Values of Quadratic Functions

(a) To complete the square for x2 kx, add 2

2

k. Then

222

22

2222

2

=

+

=

+ kxkxkxkkxx

(b) For a quadratic function y = a(x h)2 + k, (i) if a > 0, then the minimum value of y is k when x = h, (ii) if a < 0, then the maximum value of y is k when x = h.

Laws of Indices

For integral indices p and q, let a, b be real numbers. For rational indices p and q, let a, b be nonnegative numbers. 1. p q p qa a a + =

2. , where 0p

p qq

aa a

a

=

3. ( )p q pqa a= 4. ( ) p p pab a b=

5. , where 0p p

p

a a bb b

=

6. 1 , where 0pp a aa

=

7. 0 1, where 0a a=

8. ( )pq qq p pa a a= = for integers p and q where a, q > 0.

Number and Algebra

Page 5

Laws of logarithms

If , , 0 and 1, then logy ax a a x a y x= > = . For , , , 0 and , 1M N a b a b> , 1. log 1 0a = 2. log 1a a = 3. log ( ) log loga a aMN M N= + 4. log log loga a a

M M NN

=

5. log ( ) log ( is any real number)ka aM k M k= 6. loga Ma M=

7. ,logloglog

a

NN

b

ba = where b > 0 and 1b

Note: loga x is undefined for 0x .

Graphs of exponential functions and logarithmic functions

Percentages

new value initial value (1 + percenatge change)=

new value initial valuepercentage change 100%initial value

=

Selling problems A book is marked at $78 for sale. A customer is offered a 10% discount. Find the selling price. selling price = marked price (1 discount percent) $78 (1 10%) $70.2= = Simple and compound interest A P I= + simple interest %I P r n= compound interest (1 %)nI P r P= + where A: amount, P: principal, I: interest, r: interest rate per period, n: number of periods.

Ratio

Given : 3 : 4a b = and : 6 : 7b c = . Find : :a b c . Since . . . of 4 and 6 is 12L C M : : 9 :12 :14a b c =

: 3: 4: 6 : 7

: : 9 :12 :14

a bb c

a b c

=

=

=

Number and Algebra

Page 6

Variation

Direct variation , where 0y x y kx k = . Inverse Variation

1, or where 0ky y xy k k

x x = =

Joint variation 1. z varies jointly as x and y , where 0z xy z kxy k = . 2. z varies directly as x and inversely as y

, where 0x kxz z ky y

= .

Partial variation 1. z partly varies as x and partly varies as y 1 2 1 2where , 0z k x k y k k= + . 2. z partly constant and partly varies as y 1 2 2where 0z k k y k= + . 3. z partly varies as x and partly varies inversely as y

21 1 2where , 0

kz k x k k

y= + .

Errors

absolute error = measured value true value Note: maximum absolute error = largest possible error, in this case the true value is not known.

relative error absolute errortrue value

= or maximum absolute error

measured value

percentage error relative error 100%= Arithmetic sequence

, , 2 , 3 ,a a d a d a d+ + + L . Common difference d= . First term (1)T a= nth term ( ) ( 1)T n a n d= + Let the sum of the first n terms be ( ) (1) (2) (3) ( )S n T T T T n= + + + +L , then

(1) ( )( )2

T T nS n n + =

or [ ]2 ( 1)2n

a n d+ .

If , ,x y z form an arithmetic sequence, then 2

x zy += .

Geometric sequence

2 3, , , ,a ar ar ar L . Common ratio r= . First term (1)T a=

nth term 1( ) nT n ar = Let the sum of the first n terms be ( ) (1) (2) (3) ( )S n T T T T n= + + + +L , then

( 1)( ) , 11

na rS n rr

=

Sum to infinity , 1 11

ar

r= < 0, a < b then ac bc< . For c < 0, a < b then ac bc> . Graphical representation

Number and Algebra

Page 8

Solving ( ) , ( ) , ( ) and ( )f x k f x k f x k f x k> < graphically: Step 1 Draw the graph ( )y f x= and y k= . Intersecting points should be labeled. Solve 2 3 2 6x x+ + > Step 2 Identify of x which ( )f x k> . 4 or 1x x< > .

Note: > is different from . If the equation is changed to ( )f x k , then the solution is 4 or 1x x .

Solving a Linear Inequality in Two Unknowns Graphically

Linear programming (a) Solve graphically the system of inequalities:

2 23

0

x yx y

x

+

+

(b) Hence find the maximum and minimum value of ( , ) 2f x y x y= subject to the above constraints. Consider the intersecting points

(x,y) ( , ) 2f x y x y= (0,1) 2 (0,3) 6

(4, 1) 6 Maximum value = 6 and minimum value = 6 Alternative method: Draw a dotted line of 2 0x y = .

To determine the extremes of 2x y , move the dotted line vertically and find the two critical positions such that the line is about to leave the shaded region. The value of x 2y at the two extreme positions give the maximum and minimum.

Number and Algebra

Page 9

Different Numeral Systems

Numeral in the systems Numeral system Numerals

Denary system 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9 Binary system 0 and 1

Hexadecimal system 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E and F In a number, the position of each didgit has fixed place value. e.g. In 430510, the place value of 3 is 103. In C8A16, th epace value of 8 is 16. The value of a number can be expressed in the expanded form. e.g. 3 2 1104305 4 10 3 10 0 10 5 1= + + +

4 3 2 1211101 1 2 1 2 1 2 0 2 1 1= + + + +

2 1168 12 16 8 16 10 1C A = + +

Converting a binary number or a hexadecimal number into a denary number

4 3 2 1211101 1 2 1 2 1 2 0 2 1 1 29= + + + + =

2 1168 12 16 8 16 10 1 3210C A = + + =

Converting a denary number into a binary number or a hexadecimal number. 10 229 11101= 10 163210 8C A=

2 29 remainder2 14 12 7 02 3 12 1 1 0 1

LL

LL

LL

LL

LL

16 3210 remainder16 200 10 ( )16 12 816 0 12 ( )

A

C

LL

LL

LL

Transformations of functions

The following table summarizes the transformations of the graph of a function ( )y f x= and the corresponding transformations of the function.

Transformation of the graph Transformation of the function

Translate upwards by k units ( )y f x k= + Translate downwards by k units ( )y f x k= = Translate leftwards by k units ( )y f x k= +

Translate rughtwards by k units ( )y f x k= Reflect about the x-axis ( )y f x= Reflect about the y-axis ( )y f x=

Enlarge along the y-axis k times the original, where 1k > ( )y kf x=

Reduce along the y-axis k times the original, where 0 1k< <

Enlarge along the x-axis 1k

times the original, where 1k > ( )y f kx=

Reduce along the x-axis 1k

times the original, where 0 1k< <

A, B, C, D, E and F represents 10, 11, 12, 13, 14 and 15 respectively.

Measures, Shape and Space

Page 10

Mensuration of common plane figures and solids

Plane figures Perimeter Area

Trapezium

a b c d+ + + ( )2

a b h+

Rhombus

4a 2xy

Circle

2 rpi 2rpi

Sector

o

2 2360

rr

pi + 2

o360r

pi

Solids Volume Surface Area

Cube

3a 26a

Cuboid

blh 2( )bh bl hl+ +

b

a

c d h

a

a a

a

x

y a

r

a

l b

h


Page 11

Prism

Ah Base perimeter 2h A +

Cylinder

2r hpi 22 2rh rpi pi+

Pyramid

13

Ah total area of lateral facesA +

Cone

213

r hpi 2rl rpi pi+

Sphere

343

rpi 24 rpi

Eulaers formula (for a convex polyhedron only) 2V E F + =

where V: number of vertices, E: number of edges, F: number of faces.

A h


Page 12

Similar plane figures and soilds

1 12 32 3

1 1 1 1 1 1 1

2 2 2 2 2 2 2

, ,

l A l V l A Vl A l V l A V

= = = =

1 2: length of figure 1, : corresponding length of figure 2l l 1 2: face area of figure 1, : corresponding face area of figure 2A l 1 2: volume of figure 1, : volume length of figure 2V l

Congruent Triangles

Properties of congruent triangles: If ABC XYZ, then (i) A = X, B = Y, C = Z, (ii) AB = XY, BC = YZ, CA = ZX. Conditions for congruent triangles: (i) SSS (ii) SAS

(iii) AAS (iv) ASA

(v) RHS

Similar Triangles

Properties of similar triangles: If ABC ~ XYZ, then (i) A = X, B = Y, C = Z, (ii)

ZXCA

YZBC

XYAB

== .

Conditions for similar triangles:

(i) AAA (ii) 3 sides prop.

(iii) ratio of 2 sides, inc.

ZXCA

YZBC

XYAB

==


Page 13

Transformation and Symmetry in 2-D figures Reflectional symmetry and rotational symmetry

order of rotational symmetry = 3 3-fold rotational symmetry

Transformation of 2-D figures Translation Reflection

Rotation Enlargement

Geometry

Abbreviation Meaning Abbreviation Meaning

adj. s on st. line

a + 47o = 180o

s at a pt.

b + 75o + 135o = 360o

vert. opp. s

c = 50o

corr. s, AB // CD

a = 40o

alt. s, AB // CD

b = 50o

int. s, AB // CD

c + 150o = 180o

Centre of rotation

axis of relection


Page 14

sum of

=++ 180cba

ext. of

bac +=

base s isos.

If AB = AC, then CB =

sides opp. eq. s

If A = C. then AB BC= .

prop. of isos.

If AB = AC and one of the following conditions (i) BD = DC (ii) CADBAD = (iii) BCAD

prop. of equil.

If AB = BC = AC, then === 60CBA

Pyth. theorem

In ABC, if = 90C , then

222 cba =+

Converse of Pyth. thm.

In ABC, if 222 cba =+ , then

= 90C

Mid-pt. Thm.

If AM = MB and AN = NC, then MN // BC

and 12

MN BC=

Intercept Thm.

If AB// CD // EF, then : :BD DF AC CE= .

sum of polygon

Sum of all interior angles of n-sided

polygon = o( 2) 180n

sum of ext. s of polygon

Sum of all exterior angles of n-sided

convex polygon = 360o

prop. of trapezium

o180P R + = o180Q S + =

opp. sides equal

AD = BC and AB = DC


Page 15

opp. s of //gram

A = C and B = D

diag. of //gram

AO = CO and BO = DO

prop. of rhombus

prop. of //gram diagonals bisect interior angles diagonals are

prop. of rectangle

prop. of // gram equal diagonals

prop. of square

prop. of rhombus prop of rectangle

line from centre chord bisects chord

If ABON , then AN = BN.

line joining centre to mid-pt. of chord chord

If AN = BN, then ABON

equal chords, equidistant from centre

If ABOM , CDON and AB =

CD, then OM = ON.

chords equidistant from centre are equal

If ABOM , CDON and OM =

ON, then AB = CD.

at centre twice at

ce

q = 2p

in semi-circle

If AB is a diameter, then APB = 90.

converse of in semi-circle

If APB = 90, then AB is a diameter.

s in the same segment

The angles in the same segment of a

circle are equal. i.e. x = y

equal s, equal arcs

If AOB = COD,

then )

AB =)

CD


Page 16

equal s, equal chords

If AOB = COD, then AB = CD

equal arcs, equal s

If )

AB =)

CD , then AOB = COD

equal arcs, equal chords

If )

AB =)

CD , then AB = CD

equal chords, equal s

If AB = CD, then AOB = COD

equal chords, equal arcs

If AB = CD, then )

AB =)

CD

arcs prop. to s at centre

)AB :

)CD = x : y

arcs prop. to s at ce

)AB :

)CD = m : n

opp. s, cyclic quad.

A + C = 180 and B + D = 180

ext.. , cyclic quad

DCE = A

Converse of s in the same segment

If p = q, then A, B, C and D are concyclic.

opp. s supp.

If a + c = 180o (or o180b d+ = ), then A,

B, C and D are concyclic.

ext. = int. opp.

If p = q, then A, B, C and D are concyclic.

tangentradius

If PQ is the tangent to the circle at T, then

PQOT.

converse of tangentradius

If PQOT, then PQ is the tangent to the

circle at T.


Page 17

tangent properties

If two tangents, TP and TQ, are drawn to a circle from an external point T and touch the circle at P and Q respectively, then (i) TP = TQ, (ii) POT = QOT, (iii) PTO = QTO.

in alt. segment

A tangent-chord angle of a circle is equal to

an angle in the alternate segment.

ATQ = ABT

converse of in alt. segment

If ATQ = ABT, then PQ is the tangent to the circle at T.

Special lines in a triangle AD is the median of BC in ABC. CD is the angle bisector of ACB in ABC

.

CD is the altitude of AB in ABC. MN is the perpendicular bisector of AC in ABC.

Special points of a triangle Intersecting point of 3 medians Intersecting point of 3 angle bisectors Centroid Incentre


Page 18

Intersecting point of 3 altitudes Intersecting point of 3 bisectors Orthocentre Circumcentre

1. These 4 points exist for any triangle. But they may or may not coincide. 2. There exists exactly one circle with incentre as centre such that it touches all the 3 sides of

the triangle. (i.e. the 3 sides are tangents.) 3. There exists exactly one circle with circumference as centre such that it passes through all the

3 vertices of the triangle.

Coordinates Geometry

Let the points A, B be 1 1( , )x y and 2 2( , )x y respectively. Distance formul,

2 21 2 1 2( ) ( )AB x x y y= +

midpoint of AB 1 2 1 2,2 2

x x y y+ + =

If : :AP PB m n= , then 1 2 1 2,nx mx ny myPm n m n

+ + = + +

Straight lines

Slope of AB 2 12 1

y yx x

=

Let 1 2,m m be slopes of lines L1, L2 respectively and L1, L2 are not vertical lines. Then 1 2 1 2// ,L L m m= , 1 2 1 2, 1L L m m = .

Equation of straight lines Point slope form

1 1( )y y m x x = General form 0Ax By C+ + =

Slope AB

= if 0B (if B = 0, is undefined)

xintercept CA

= if 0A (if A = 0, the line does not cut the xaxis)

yintercept CB

= if 0B (if B = 0, the linen does not cut the yaxis)


Page 19

Equation of a circle Let (h,k) be the centre and r be the length of the radius. Centreradius form

2 2 2( ) ( )x h y k r + = General form

2 2 0x y Dx Ey F+ + + + =

Centre ,2 2D E

=

Radius 2 2

2 2D E F +

Polar coordinates

Polar coordinates of P = ( , )r , r OP= , tan slope of OP = Trigonometric ratios of an acute angle

hypotenuseside opposite

sin =

hypotenusesideadjacent

cos =

sideadjacent side opposite

tan =

Trigonometric ratios any angle Sign of trigomometric ratios

Trigonometric identities

1. sintancos

= 2. 2 2sin cos 1 + =

3. osin(90 ) cos = 4. ocos(90 ) sin =

5. o 1tan(90 )tan

= 6. osin(90 ) cos + =

7. ocos(90 ) sin + = 8. o 1tan(90 )tan

+ =


Page 20

9. sin(180 ) sin = 10. cos(180 ) cos = 11. tan(180 ) tan = 12. sin(180 ) sin + =

13. cos(180 ) cos + =

14. tan(180 ) tan + = 15. sin( ) sin(360 ) sin = =

16. cos( ) cos(360 ) cos = = 17. tan( ) tan(360 ) tan = = 18. sin(360 ) sin + =

19. cos(360 ) cos + = 20. tan(360 ) tan + = Graphs of trigonometric functions Graphs of y = sinx

Graphs of y = cosx

Graphs of y = tanx


Page 21

Application of Trigonometry

Sine Formula

sin sin sina b c

A B C= =

Cosine Formula

2 2 2 2 cosc a b ab C= + Area of a triangle Area 1 sin

2ab C=

Herons Formula

Area ( )( )( )s s a s b s c= where 2

a b cs

+ += .

Angle between a line and a plane A. Projection on a plane (a) Projection of a point on a plane (b) Projection of a line on a plane

If PQ is perpendicular to any straight line If AB intersects pi at A, and C is the (L1 and L2) on pi passing through Q, then projection of B on pi, then (i) PQ is perpendicular to pi, (i) BC is perpendicular to pi, (ii) Q is the projection of P on pi, (ii) AC is projection of AB on pi, (iii) PQ is the shortest distance between (iii) BC is the distance between B and pi. P and pi.

B. Angles between lines and planes (a) Angle between two intersecting straight (b) Angle between a straight line and a lines. plane.

is the acute angle formed by the two is the acute angle between the straight lines. It is called the angle of intersection. line and its projection on the plane.

(c) Angle between two intersecting planes (i) AB is the line of intersection of pi1 and pi2. (ii) is the acute angle between two interescting straight lines on the planes, and which are perpendicular to AB.

C. Line of greatest slope Lines of greatest slope

(a) AB is the line of intersection of the planes ABCD and ABEF.

(b) PX, L1, L2 and L3 are perpendicular to AB. They are lines of greatest slope of the inclined plane ABEF. Angle of elevation and angle of depression

Data Handling

Page 22

Simple Statistical Diagrams and Graphs

Pie chart Stem-and-leaf diagram e.g.

e.g.

The data presented above are: Angle of the sector representing the 15 kg, 16 kg, 21 kg, 21 kg, 27 kg, colour Blue = == 144%40360x . 33 kg, 34 kg, 36 kg, 42 kg, 51 kg

Histograms e.g. The histogram on the right is constructed

from the frequency distribution table below.

Hourly wage

($)

Class boundaries

($)

Class mark

($)

Frequency

25 29 24.5 29.5 27 15 30 34 29.5 34.5 32 22 35 39 34.5 39.5 37 11 40 44 39.5 44.5 42 4 45 49 44.5 49.5 47 3

Frequency polygon e.g. The following table shows the amount of time spent by 60 customers in a supermarket.

Time spent (min)

Class mark (min)

No. of customers

11 20 15.5 9

21 30 25.5 21

31 40 35.5 16

41 50 45.5 8

51 60 55.5 4

61 70 65.5 2

Weights of 10 dogs

Stem (10 kg) leaf (1 kg) 1 5 6 2 1 1 7 3 3 4 6 4 2 5 1

Data Handling

Page 23

Cumulative frequency polygon The following table shows the age distribution of 50 teachers in a school

Age 20 24 25 29 30 34 35 39 40 44 45 49 Frequency 2 9 18 11 6 4

The cumulative frequency table of the above data is constructed below

Age less than 19.5 24.5 29.5 34.5 39.5 44.5 49.5

Cumulative Frequency 0 2 11 29 40 46 50

The cumulative frequency polygon of the above data is constructed below

Measures of central tendency

Mean

Ungrouped mean 1 2 3 nx x x xxn

+ + + +=

L

Grouped mean 1 1 2 2 3 31 2 3

n n

n

f x f x f x f xx f f f f

+ + + +=

+ + + +

L

L

Weighted mean 1 1 2 2 3 31 2 3

n n

n

w x w x w x w xx

w w w w

+ + + +=

+ + + +

L

L

Median For ungrouped data, if number of data, n, is: (i) odd, median = middle datum (ii) even, median = mean of the 2 middle data

For data grouped into classes, the median can be determined from its cumulative frequency polygon/curve. e.g. The cumulative frequency polygon on the right

shows the donations by a group of students to a charity fund. From the graph, the number of students is 100. From the graph, the median donation by the group of students is $52.

Mode The data item with the highest frequency. Model class The class with the highest frequency

Data Handling

Page 24

Measures of Dispersion

Range Ungrouped data: range = largest datum smallest datum Grouped data: range = highest class boundary lowest class boundary. Inter-quartile range Ungrouped data:

inter-quartile range = upper quartile (Q3) lower quartile (Q1) Grouped data: The quartiles can be read from the corresponding cumulative frequency polygon (or curve). inter-quartile range = upper quartile (Q3) lower quartile (Q1). Box-and-whisker diagram The box-and-whisker diagram is an effective way to present the lower quartile, the upper quartile, the median, the maximum and the minimum values about a data set

Standard deviation

2 2 2 21 2 3( ) ( ) ( ) ( )nx x x x x x x x

n

+ + + + =

L

For grouped data,

2 2 2 21 1 2 2 3 3

1 2 3

( ) ( ) ( ) ( )n nn

f x x f x x f x x f x xf f f f

+ + + + =

+ + + +

L

L

Standard Score For a set of data with mean x and standard deviation , the standard score z of a given datum x is

given by

xxz

= .

Normal distribution Characteristics of a normal curve: 1. It is bell-shaped. 2. It has reflectional symmetry about xx = . 3. The mean, median and mode are all equal to x .

Data Handling

Page 25

Probability

Theoretical probability In the case of an activity where all the possible outcomes are equally likely, the probability of an event E, denoted by P(E), is given by

Theoretical probability

outcomes possible ofnumber totalevent the tofavourable outcomes ofnumber )( EEP =

where 1)(0 EP . Note that P(impossible event) = 0 and P(certain event) = 1

Experimental probability The experimental probability P(E) of an event E is defined as follows:

trialsofnumber totalexperimentan in occurs event timesofnumber )( EEP = or

EEP event offrequency relative)( = For a large number of trials, the experimental probability is close to the theoretical probability which is deduced from theories. Permutation An arrangement of a set of objects selected from a group in a definite order is called a permutation. (i) The number of permutations of n distinct objects is n!.

e.g. Arrange 5 students in a row. Number of arrangements = 5! = 120

(ii) The number of permutations of n distinct objects taken r at a time, denoted by nrP , is )!(

!rn

n

.

e.g. Choose 5 out of 7 students, and arrange them in a row. Number of arrangements 252075 == P Combination A selection of r objects from n distinct objects regardless of their order is called a combination. The number of combinations of n distinct objects taken r at a time, denoted by nrC , is !)!(

!rrn

n

.

e.g. Choose 2 from a list of 12 friends to invite to a party. Number of ways of choosing 66122 == C

Additional law of probability Mutually exclusive events events that cannot occur at the same time.. If events A and B are mutually exclusive, then

( ) ( ) ( )P A B P A P B = + If events 1 2 3, , , , nE E E EL are mutually exclusive events, then

1 2 3 1 2 3( ) ( ) ( ) ( ) ( )n nP E E E E P E P E P E P E = + + + +L L If events A and B are not mutually exclusive, then

( ) ( ) ( ) ( )P A B P A P B P A B = + where ( ) 0P A B Multiplication Law of Probability If the probability that one event occurs is not affected by the occurrence of another event,

these two events are called independent events. Multiplication law of probability for independent events: If A and B are two independent events, then )()()( BPAPBAP =

If the probability that one event occurs is affected by the occurrence of another event, these two events are called dependent events.

Data Handling

Page 26

Multiplication law of probability for dependent events: If A and B are two dependent events, then ( ) ( ) ( | )P A B P A P B A = , where

( | )P B A is the probability that event B occurs given that event A has occurred. Expected value (i.e expectation)

1 1 2 2 3 3( ) n nE X Px P x P x P x= + + + +L , where 1 2, ,P P L are probabilities of events 1 2, ,E E L and 1 2, ,x x L are the corresponding value of 1 2, ,E E L .

Complementary events If A and A are complementary events, then (1) A happens if A does not happen and B does not happen if A happens; (2) ( ') 0P A A = (i.e. mutually exclusive); (3) ( ') 1 ( )P A P A=

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