Permutation & Combination

What is Permutation?

The number of different ways that a certain number of objects can be arranged in order from a large name of objects. Below are the hints when you encounter a problem :

(1) Order matters, An order list.
(2) Arrangement.

What is Combination?

The number of different ways that a certain number of objects as a group can be selected from a layer number of objects. Below are the keys when you encounter a problem :

(1) Group
(2) Sample
(3) Choice
(4) Election
(5) Collection

Ways:
1! = 1 way
2! = 2 ways
3! = 6 ways
4! = 24 ways
5! = 120 ways

Symbols:
N! = Permutation
! = Factorial
n = number selected

The Formula?

Example & Questions :

Suppose there are 12 people and only 8 people are invited to a party. In how many ways can the party be selected?

Solution :

Answer of  Example 1 ( Combination )

Example 2 :

Answer of example 2 (Permutation)

Probability

Probability

What is Probability, Experiment, Outcome and Sample Space?



Probability is the measure of how many an event is

Experiment is a situation involving chance or probability

Outcome / Sample is the result of a single trial of an experiment

Sample Space / Event is all the possible outcome of an experiment (s) sample points

Example: "It is unlikely to rain tomorrow" :

Examples

The Formula?

Formula of Probability

Examples & Questions :-

Probability of a card (1st example)

A card is chosen at random from a deck of 48 playing cards.

There are 4 Queens and 4 Kings in a deck of playing cards.

What is the probability the card chosen is a Queen or a King?

The Answer is = 1/6

Solution

There are 4 Queens and 4 Kings, so the Number of ways it can happen = 8
There are 52 cards altogether, so the Total number of outcomes = 48


So the Probability the card of either Queens or King is

Examples 2

Alex has a bag with 9 blue sweets and 3 red sweets in it. She picks up a sweet at random from the bag, replaces it and then picks again at random. Draw a tree diagram to represent this situation and use it to calculate the probabilities that she picks:

(a) two red sweets

(b) no red sweets

(c) at least one blue sweet

(d) one sweet of each color


Solutions :

the probabilities that she picks:

 (a) two red sweets
 

(b) no red sweets

(c) at least one blue sweet

(d) one sweet of each color

Measure of Dispersion

Title

what is Measure of Dispersion?

A measure of Dispersion is a method of measuring the degree by which Numerical data or values tend to speed from or cluster about central point of average.


The common Measure of Dispersion
are the following:

(1) The range
(2) The Quartile Deviation
(3) The Standard deviation and Variance

The Range

There are two formula of using range which are:

Grouped Data:

Example:

Following are the wages of 9 workers of an Engineer. Find the range. Wages in Dollars 14270, 1450, 1555, 1280, 1496, 1495, 1590, 1488.

R = 1590 - 1280
   Ans = 310

Solution:

First step : Find the largest and Smallest value first

Second step : Once you find those two numbers, then you have to minus those two numbers together and you will find the answer.


 Ungrouped of Data :

 Example:

 Calculate the range.

Solution:

The Upper class boundaries of the highest class is = 74.5 
The Lower class boundaries of the highest class is = 59.5  

          
                           Range = 75.5 - 59.5
                                    Ans  = 15 kg

First step : Create another 2 coloumns which are Class Boundaries and Mid Value

Second step : Create a new record every each rows like above (59.5 and 62.5) depends on what value  are given.

First step : Find the largest and Smallest value at Class Boundaries coloumns

Second step : Once you find those two numbers(Largest & Smallest), then you have to minus those two numbers together and you will find the answer.

Note* Using these two formula of Range, first you should understand the question first which formula you use. Sometimes they give the question either Grouped data or ungrouped of Data.

The Quartile Deviation

what is Quartile deviation?

Quartile Deviation (QD) means the semi variation between the upper quartiles (Q3) and lower quartiles (Q1) in a distribution. Q3 - Q1 is referred as the interquartile range.

Formula?

QD = Q3 - Q1 / 2

Example:

Calculate the QD for a group of data, 267,544,440,255,310,387,840,966.

Solution:

Given data = { 267,544,440,255,310,387,840,966 }

Step 1 :

First, arrange the given digits in ascending order 255,267,310,387,440,544,840,966


Step 2 :

From the given group of data, { 255,267,310,387,440,544,840,966 } Consider,
First four values Q1 = 255,267,310,387
Last four values Q3 = 440,544,840,966


Step 3 :

Now lets find the median value for Q1
(Q1) = 267+310/2
(Q1) = 577/2 = 288.5

Step 4 :

Let us now calculate the median value for Q3
(Q3) = 544+840/2
(Q3) = 1384/2 = 692

Step 5 :

Now, find the median value between Q3 and Q1.
Quartile Deviation = Q3-Q1/2
= 692 - 288.5/2
= 403.5/2
= 201.75

The Standard Deviation & Variance

 
What is Standard Deviation?

The Standard Deviation is a measure of how spread out value are.

What is Variance?

The average of the squared differences from the Mean.


Formula?

Formula of Standard Deviation

∑ = Summation for

X = X far

N = Sample size

U = Mean

Example:

A difference math class took the same test with these five scores {79,83,83,65,54}
Calculate the Variance and Standard Deviation

Solution:

Step 1 :

Find the mean  :

Mean  =  83 +83 + 79 + 65 + 54  = 364/5  =  72.8

So the mean average is 182. Then lets moving on to the 2nd step.

Step 2 :

To find the Variance, take each scores, square it, and then average the result:

⁼ (79 – 72.8)² + (83 – 72.8)² + (83 – 72.8)² + (65 – 72.8)² + (54 – 72.8)²

= (38.44) + (104.04) + (104.04) + (60.84) + (353.44)

= 660.8 (Variance)

 

Step 3 :

To find the Standard Deviation, just square root of variance.

= 25.71 (Standard Deviation)