Probability (cont.)

TREE DIAGRAMS

  §         Independent events and their probabilities can be shown on a tree diagram.  Each event is represented by a branch

  §        E.g.  A coin is flipped twice. Draw a tree diagram to show all the possible outcomes

EXAMPLE:

Measure Of Dispersion (Cont.)

  3.    The standard deviation

• A deviation is the difference between an individual value in a set of data and the mean for the data.
• Standard Deviation averages the square of the distance that each piece of data is from the mean.
• The smaller the standard deviation, the more compact the data set.

                                                          Standard Deviation – Population

                                                         
                                                                 Standard Deviation – Sample

    Measure Of Dispersion (Cont.)

  2.    The Quartile Deviation

      §Quartiles divide a set of ordered data into four groups with equal numbers of values

The three “dividing points” are the first quartile (Q₁), median, (sometimes called the second quartile, Q₂), and the third quartile (Q₃)

§  The interquartile range is Q₁ – Q₃, which is the range of the middle of the data.

§  The semi-interquartile range is one half of the interquartile range.

§  Both these ranges indicate how closely the data are clustered around the median.

The quartile deviation is called the semi-quartile range IT is defined as the amount of dispersion present in the middle of 50% of the values, hence the equation is given by,

Q.D = Q3 – Q1            2

Q.D = the quartile deviation

Q1 = the first quartile

Q2 = the third quartile

                                                

Q1 = L + (N/4 – F1) i              F

Where:

Q = the first quartile

L = the lower limit of the first quartile class (i.e. the class containing the N/4^th item)

            Strictly, L is the lower boundaries of the 1^st quartile class.

F1 = the cumulative frequency just above the 1^st quartile class.

F = the frequency of the 1st quartile class

N = the total frequency

I = the class interval

Class Intervals	Frequency	M	Quartile class	Cf>
45 – 49	3	47		3
50 – 54	4	52		7
55 – 59	6	57	Q1	13
60 – 64	7	62		20
65 – 69	10	67		30
70 – 74	7	72		37
75 – 79	6	77		43
80 – 84	4	82		47
85 – 89	3	87		50

Solution:

Q1 = L + (N/4 – F1) i            F N/4 = 50/4 = 12.5 Q1 = 54.5 (12.7 – 7)5               6 Q1 = 59.08

3^rd Quartile:

Formula :

                                   

Q3 = L + (3N/4 – f1) i              F

Where :

F3  =  the cumulative frequency just above the 3^rd quartile class.

Example :

Q3 = L + (3N/4 – f1) i                       F 3N/4 = 150/4 = 37.5 Q3 = 74.5 + (37.5 – 37) 5                              6             74.5 + (O.O833) 5             74.5 + O.4165             74.9165             =74.92 Q.D = Q3 – Q1/2 = 74.92 – 59.08/2 = 7.92

      

Introduction of statistical data (Cont.)

o   Discrete – data can only take on certain individual values.

§  Example:       

·         Number of pages in a book is a discrete data.

·         Shoe size is a Discrete data.  E.g.  5, 5½, 6, 6½ etc.  Not in between.

·         Number of people in a race is a discrete data.

o   Continuous data – data can take on any value in a certain range.

§  Example:

·         Length of a film is a continuous data.

·         Temperature is a continuous data.

·         Time taken to run a race is a continuous data.

Sample – The subcollection data drawn from the population.

Population – The complete collection of all data to be studied.  

SET THEROY

Set – is the collection of distinct objects

Example:

- Set Symbol -

Permutation & Combination

PERMUTATION

The number of different ways that a certain number of objects can be arranged in order from a large number of objects.

Ordered list: Order Matters

Keyword: Arrangements

Formula:             N!

            (N-n)!

Example:

How many ways can you organize those 5 objects?

COMBINATION

The number of different ways that a certain number of objects as a group can be selected from a larger number of objects.

Unordered group/set: Order does not matters

Keyword(s): Choice, selection, election

Formula:                           N!

                  n!(N-n)!

 Example:

How many ways can you organize those 5 objects picking 3 at a time?

Probability

      ·         Probabilities are written as fractions or decimals, and less often as percentages

      ·         An event can have several possible outcomes

      ·         Each outcome has a probability or chance of occurring

      ·         When a fair dice is thrown there is equal chance of throwing each number.  The outcomes from the event throwing a dice are equally likely outcomes

      ·         If the outcomes of an event are equally likely, the probability can be calculated using:

Probability of an event = number of successful outcomes

                                                Total number of possible outcomes

MUTUALLY EXCLUSIVE EVENTS

      ·         Outcomes that cannot happen at the same time are called mutually exclusive outcomes

      ·         E.g. A dice is rolled.  It shows 5.  It is rolled again.  It shows 2. These events cannot happen at the same time.  They are mutually exclusive

      ·         The total probability of mutually exclusive outcomes is 1.  An event cannot happen and not happen at the same time

      ·         The sum of the probabilities of mutually exclusive outcomes is 1

      ·         Probability of rolling a 5

= 1 – Probability of NOT rolling a 5.

THE “OR” RULE

       ·         If two events, A and B are mutually exclusive:

            ·         P(A or B) = P(A) + P(B)

       ·         This is known as the “OR” rule or addition rule for mutually exclusive probabilities.

INDEPENDENT EVENT

  ü  Two events are independent when the probability of one event happening is not affected by the outcome of the other event

  ü E.g.  Roll a dice and flip a coin.

Event A:  the dice shows an odd number

Event B:  the coin shows tails.

These events are independent.  Neither outcome can influence the other.

  ü Are these events independent?

You go outside.  Event A:  It is snowing Event B:  It is cold

THE “AND” RULE

  §        To find the probability of two independent events both happening, multiply the individual probabilities together

  §         If A and B are individual events

P(A and B) = P(A) x P(B)

This is the AND rule or multiplication rule.

E.g.  The probability of rolling an odd number on a dice AND flipping a coin to get tails is:

P(Odd) x P(Tails) = ½ x ½ = ¼

Probability (cont.)