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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
Measures, Shape and Space
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
Measures, Shape and Space
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
==
Measures, Shape and Space
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
Measures, Shape and Space
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
Measures, Shape and Space
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
Measures, Shape and Space
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.
Measures, Shape and Space
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
Measures, Shape and Space
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)
Measures, Shape and Space
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
+ =
Measures, Shape and Space
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
Measures, Shape and Space
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=