unit 6

Statistics and probability

Chapter 15 & 16

Collecting data

When there are too many numbers reaching to thousands, we are not able to manage them well so we do collecting data. By applying statistics properly we can highlight the important aspects of data and make the data easier to interpret. 

Quantitative data

Data that can be written as numbers

Qualitative data

Data can't be written as numbers

Grouping data

Subdividing the full range of values into a few sub-ranges. By assigning each continuous value to the sub-range or class within which it falls, the data set changes from continuous to discrete.

Example :Groups and histograms

The heights in centimetres of 30 learners are given below.

142	163	169	132	139	140	152	168	139	150
161	132	162	172	146	152	150	132	157	133
141	170	156	155	169	138	142	160	164	168

Group the data into the following ranges and draw a histogram of the grouped data:

130140150160170≤h<140≤h<150≤h<160≤h<170≤h<180

Answer:

56	49	40	11	33	33	37	29	30	59
21	16	38	44	38	52	22	24	30	34
42	15	48	33	51	44	33	17	19	44
47	23	27	47	13	25	53	57	28	23
36	35	40	23	45	39	32	58	22	40

Probability

P(E) = The number of ways that event can occur n(E)      
           The number of elements in the sample space n(S)

P(E) = n(E)
           n(S)

                       Probability is always between 0 and 1

               The probability is on the branch and the outcome is at the end of the branch.

CONSECUTIVE EVENTS
Consecutive events are those events which occur one after another, we can find the probability of these types of events by multiplying the probability of one consecutive event with other. 

Example
A die is rolled into the surface and a coin is tossed up in the air and a card is drawn from a pack of 52 cards consecutively. So find consecutive probability of getting 5 on die, head on coin and ace on card.
Solution: firstly we will find the Probabilities separately then we will combine them,
Probability of getting 5 = 5/6,
Probability of getting a head = 1/2
Probability of getting a ace = 4/52=1/13
Result: = 5/6*1/2*1/13
            = 5/156

INDEPENDENT EVENTS
Two events, A and B, are independent if the fact that A occurs does not affect the probability of B occurring.

Examples:

• Landing on heads after tossing a coin AND rolling a 5 on a single 6-sided die.
• Choosing a marble from a jar AND landing on heads after tossing a coin.
• Choosing a 3 from a deck of cards, replacing it, AND then choosing an ace as the second card.

DEPENDENT EVENTS

Two events are dependent if the outcome or occurrence of the first affects the outcome or occurrence of the second so that the probability is changed.

Example:
 A card is chosen at random from a standard deck of 52 playing cards. Without replacing it, a second card is chosen. What is the probability that the first card chosen is a queen and the second card chosen is a jack?

Answer: 
The probability that the first card is a queen is 4 out of 52. However, if the first card is not replaced, then the second card is chosen from only 51 cards. Accordingly, the probability that the second card is a jack given that the first card is a queen is 4 out of 51.

Unit 4

Further Geometry

Chapter 10,11 & 13

Geometry
= the branch of mathematics concerned with the properties and relations of points, lines, surfaces, solids, and higher dimensional analogs.


SURFACE AREA
The total area of the surface of a three-dimensional object.

CUBE 

: 6 a ²

 (a is the length of the side of each edge of the cube) 

RECTANGLE

: 2ab + 2bc + 2ac

 (a, b, and c are the lengths of the 3 sides)

PRISM

: Lateral area + Area of two ends

: (perimeter of shape b) * L+ 2*(Area of shape b)

  (b is the shape of the ends)

SPHERE

: 4 pi r ²

 (r is radius of circle)

CYLINDER

:  2 pi r ² + 2 pi r h

 (h is the height of the cylinder, r is the radius of the top)

SIMILARITY

• Two figures are similar when one figure can be enlarged and superimposed so they exactly coincide
• They have the same shape, but different  size

CIRCLE GEOMETRY
It is the set of all points in a plane that are at a given distance from a given point, the centre; equivalently it is the curve traced out by a point that moves so that its distance from a given point is constant.

Points

Ø The basic geometric figure is a point.

Ø Use points to specify exact location.

Lines

Ø A line is 1Dimensional, having length but no width or height.

Ø Lines are determined by 2 points

Planes

Ø 2dimensional flat surface, having length and width but without height.

Ø It extends on all sides, composed with an infinite number of points and lines

Intersection lines

Ø Lines or line segments that meet or cross at a certain point

Angles

Ø When 2 lines meet at a point (the end), they will make a certain angle.

Ø Every angles are not always the same

• In a circle, a radius perpendicular to a chord bisects the chord.
• a radius that bisects a chord is perpendicular to the chord.
• the perpendicular bisector of a chord passes through the center of the circle.

Intersecting Chords Rule

: (segment piece)×(segment piece) = (segment piece)×(segment piece)

Secant-secant rule:

: (whole secant)×(external part) = (whole secant)×(external part)

Secant-tangent rule

: (whole secant)×(external part) = (tangent)2

ANGLE PROPERTIES 

 The angle at the center is twice bigger of the angle at the circumference.

Both angles are in the same size.

 It's 90 degree, if we prove using the theory above.

Resources:

http://www.mathgoodies.com/lessons/vol2/geometry.html

https://www.mathsisfun.com/definitions/circle.html

http://www.math.com/tables/geometry/surfareas.htm

http://mathschallenge.net/library/geometry/angle_properties