Exhaustive Events

Introduction to exhaustive events:

Random Experiment:

Random experiment is an experiment in which outcome is not known in advance. For example, if a uniform unbiased coin is tossed, then the outcome may be tail or head.

Exhaustive Events:

All possible outcome of an experiment is called exhaustive events or exhaustive cases. In tossing a coin, either the head or tail turns up. There is no other possibility and therefore these are the exhaustive events.Let us see concepts and sample problems for exhaustive events. Two or more events that have the property that their union equals the sample space is called exhaustive event.

Exhaustive events covers all the possible outcomes that is sample space of the problem.


Exhaustive events:


Exhaustive events:

If two or more events together form sample space S then these events are said to be exhaustive events. I

Example 1:

In an experiment of throwing the die find the chance of exhaustive events.

Solution:

" n " throwing a die,

The events of getting an odd number and the event of getting an even number together form the sample space.

A = {The events of getting an odd number} = {1,3,5}


B = {The events of getting an even number} = {2,4,6}

A ? B = {1,2,3,4,5,6} = {S}

So, they are exhaustive events.

Example 2:

In an experiment of tossing three coins, consider the following events.

A : exactly one head appears,

B : exactly two heads appear

C : exactly three heads appear

D : atleast two tails appear

S = {HHH, HHT, HTH, THH, HTT, THT, TTH, TTT}

A = {HTT, THT, TTH}

B = {HHT, HTH, THH}

C = {HHH}

D = { TTH, THT, HTT, TTT}

The events A, B, C and D together form the sample space S.

That is, S = A ? B ? C ? D. Therefore A, B, C and D are called exhaustive events.


Exhaustive events:


Example 3:

In an experiment of tossing two coins, consider the following events.

A : exactly one head appears,

B : exactly two heads appear

C : exactly two tails appear

S = {HH, HT, TH, TT}

A = {HT, TH}

B = {HH}

C = {TT}

The events A, B and C together form the sample space S.

That is, S = A ? B ? C. Therefore A, B and C are called exhaustive events.

Example 4:

In an experiment of card game find the chance of exhaustive events.

Solution:

" n " getting a card,

The events of getting an diamond card,event of getting an heartine card,event of getting an spade,event of getting an club card together form the sample space (S).

Sample space(S) = 54.

A = {The events of getting an diamond card } = {13}

B = {The events of getting an heartine card }  = {13}

C = {The events of getting an club card }        = {13}

D = {The events of getting an spade card }     = {13}

A ? B ? C ? D = {54} = {S}

So, they are exhaustive events.

Function Substitution

Introduction to function substitution:

In mathematics, function substitution system is a method used to solve system of linear equations. A system of linear equation is a position of two or three linear equations by means of two or three variables in that order.

In the direction of answer system of linear equations, we contain to solve one of those equations for an exacting variable. The explain equation is now substituted in the other equation, and the value of the other variable is obtained. In this article we are going to study how to do function substitution to solve system of linear equations.

Substitution method:

For answer the systems of equations, function Substitution is used.

In the function substitution method of solving equations, from side to side a exacting variable, another variable can be solved if any one of the equations is solved.


Steps involved in solving function Substitution method:


For answer the system of linear equations using the method of function substitution, the subsequent steps are to be followed:

Step 1: Answer anyone of the equation to write one variable in terms of other variable.

Step 2: After that Substitute this in the second equation to obtain a single variable equation.

Step 3: The after that step is to answer the single variable equation to locate the value of that variable.

Step 4: Once we obtain the value of one variable, substitute the value in some of the equation to obtain the value of the second variable.

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Solving Problems based on function Substitution:


Example 1:

Answer the subsequent system of linear equations using the method of function substitution.

x - y = -5

3x+8y = -48.
Solution:

Step 1: Rearrange the first equation,
x - y = -5
y = x + 5

Step 2:Substitute this value for y into the second equation;
3x + 8(x + 5) = -48

Step 3:Expand and simplify the equation:
3x + 8x + 40 = -48
11x = -88
x = -8

Step 4:Substitute x back into one of the original equations;
-8 - y = -5
y = -3

Solution:  x = -8, y = -3

Example 2:

Answer the subsequent system of linear equations using function Substitution method:

x + y = 25

-4x + y = 10.
Solution:

Step 1: Rearrange the first equation,
x + y = 25
y = 25 - x

Step 2:Substitute this value for y into the second equation;
- 4x + (25 - x) = 10

Step 3:Expand and simplify the equation:
-4x + 25 - x  = 10
-5x = 10 - 25
-5x = -15

x = 3

Step 4: Substitute x back into one of the original equations;
3 + y = 25
y = 22

Solution: x = 3, y = 22

These are the examples for solving substitution method.

Geometric Introduction

Introduction to geometry:
Geometry is one of the main branch of mathematics it deals with the pre algebra concepts.Geometry is the study of object; It contains the 2 dimensional objects and three dimensional objects. Geometry is the study of these objects size and shapes .Each object have different sizes and different shapes. Without geometry we cannot easily find the object shapes and size of an object. Main concept of geometry is line, angles, Line segment.

Basic concepts of geometry introduction:

Line
Line segment
Angles
Rays
Introducing geometric two dimensional objects:

Square
Rectangle
Triangle
Circle
Three dimensional objects:

Cube
Cylinder
Prism
Sphere

Basic terms of geometry Introduction:


Basic terms in geometry:

Introduction for line:

Line is nothing but straight line curve; It is a infinite length of one dimensional object. It plot the doubled direction, that Mean we can draw the line both direct and inverse direction

Introduction for line segment:

Line segment mean nothing but a one-dimensional, curved finite length  of the single direction line is called line segment.

Two types of lines:

Horizontal line
Straight line is called horizontal line

Vertical line
When the lines are stand at 90 degree called as vertical line

Comparison of two lines also classified into many types:

Parallel line
When the two lines stands opposites to each other called parallel line example Ladder)

Perpendicular lines
When one vertical line stands on horizontal line that is called as Perpendicular line.(example alphabet letter “T”)

Intersection line:
When two lines are intersected each other (Example: Alphabet x)


Basic terms of angles in geometry Introduction:


Angles:

Angle mean nothing but a Inclination of two line is called angles

Angles are classified in different ways:

Right angle
Right angle mean nothing nut a angle between two lines is 90 degree

Acute angle
Acute angle mean nothing but angle between two line s is < 90 degree

Abtuse angle
Abtuse angle mean only one angle should be > the 90 degree

Straight angle:
It is nothing but Horizontal line. Angle between the line is 180 degree

Other angles using in geopmety:

Supplementary angle
Complementary angle
Vertical angle
Reflex angle

Math Mid Term Help

Introduction to math mid term help:

Math mid term help is the practice problems for math mid term exam. Mid term exam is conducted to check the performance of  students. Math mid term help covers many topics under mathematics. Below given some of examples and practice problems to math mid term help. Math lessons are used in real life situations such as counting the money, calculating time, measuring weight, measuring distance, etc. In this article math mid term help let us see practice problems and practice exam with solutions.


Example Problems - Math mid term help:


Example 1:

Calculate median of the values

55, 65, 23, 11, 77

Solution:

Arrange in ascending order

11, 23, 55, 65, 77

Median = 55

Example 2:

Find the factor of 17

Solution

We know that 17 is a prime number

All the prime numbers have only two factors

The two factors are 1 and the numbers itself

So the factor of the number 17 is 1 , 17 alone.

Example 3:

Find the area of triangle if its base is 20cm and height is 9 cm.

Solution:

The formula for area of triangle = ` (1 /2 xx base xxheight)`

Here, base = 20 cm

height = 9 cm

Therefore, area of triangle = `(1/2)xx20xx9 `

= `(1/2)xx 180`

= 90 cm^2.


Few More Example Problems - Math mid term help:

Example 4:

Evaluate  25 – 6 × 2 +6

Solution:

Here we use PEMDAS rule for solving the problem since it has various operations to perform

From this question there is no parenthesis exponents and division.

So we perform multiplication first.

6 × 2 = 12

25 -12 + 6

13 + 6

The answer is 19.

Example 5:

Find the value of a in the given

4a + 40 = 0

Solution:

Subtract 40 on both side

5a + 40 -40 = -40

5a = -40

a =-`40/5`

a = -8

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Practice problems - Math mid term help:


Below are some of the problems for practice with answers. You can work out and check with this answer

Practice midterm exam:

1. Add 256.234 + 210.325

2. Subtract 64.23 - 23.65

3. Find the median of the given term 101, 112, 108, 105, 123

4. Solve using PEMDAS rule. 12 +14 x2 - 22

Answer:

1. 466.559

2. 40.58

3. 108

4. 36

Pattern Math Terms

Introduction-pattern math terms:

A pattern, from the French patron, is a type of theme of recurring events or objects, sometimes referred to as elements of a set. These elements repeat in a predictable manner. It can be a template or model which can be used to generate things or parts of a thing, especially if the things that are created have enough in common for the underlying pattern to be inferred, in which case the things are said to exhibit the unique pattern.

Source: Wikipedia


Concept-pattern math terms:


In math, numeric patterns in algebra are pattern ready from numbers.

The numbers can be in a listing. Any math process likes addition, subtraction, multiplication, or division, you create the pattern.

Many of the patterns in math you observe will use addition. The same number will be additional to each number in the list to create the next number in the pattern math.

Example for pattern math terms:

2, 4, 6, 8, 10, 12, 14…

The pattern in math is to adding 2 always. The subsequently number is 16, then 18.


Example- pattern math terms:


Blue, black, orange and brown counters are arranged in a row. The brown counter is to the left of the orange counter and to the right of the blue counter. The black counter is to the left of the orange counter and not next to the blue one. What are the colors of the counters in the row from left to right?

Solution:

Given:

The brown counter is to the left of the orange counter and to the right of the blue counter.

Blue counter is to the left of brown counter

Brown counter is to the left of the orange counter

Given:

The black counter is to the left of the orange counter and not next to the blue one

Black counter is to the left of the orange counter

Black counter is not next to the blue counter

Black counter is between brown counter and orange counter.

So the order is: Blue, Brown, Black and Orange.

Equalities in Math

Introduction to equalities in math:

A state of two values in equal position is termed as equalities in math. The symbol ‘=’ is used for representing a equalities of values. On the other hand we can say that the ‘=’ symbol should match the right hand side to the left hand side.

Example, p=q. Here p, q are termed as equalities in math.


Equalities in math:


Equalities allow some basic operation of math.

Addition:

By adding a number on both sides of equality results another equality.

Example:

Consider an equality p+q. In this if we add the value x, then the equality becomes p+x=q+x.

Subtraction:

By subtracting a common value on both sides of equality results another equality.

Example:

Consider an equality p+q. In this if we subtract the value x, then the equality becomes p-x=q-x.

Multiplication:

If we multiply a number on both sides of equalities, then the result is same.

Example:

p.c = q.c

Division:

If we divide a number on both sides of equalities, then the result is same.

Example:

p `-:` c = q `-:` c


Example problems for equalities in math:


Example: 1

Which of the following is equal 6+10

a. 10+6

b .8+8

c. 2+10

d. 3+7

Solution:

Given equality is 6+10

6+10 = 16

The first choice 10+6 gives =16

Hence 6+10 = 10+6

Answer: 10+6

Example: 2

Which of the following is equal to 9 . 12 =

a) 15 . 2

b) 27 . 4

c) 10 . 12

d) 5.15

Solution:

Given 9 .12 = 108

First choice gives 15 . 2 =30

Second choice gives 27 . 4 =108

Answer: 27 . 4


Practice problems for equalities in math:


Problem: 1

Which of the following is equal to 144 `-:` 2

a) 158 `-:` 9

b) 162`-:` 4

c) 216 `-:` 3

d) 404 `-:` 10

Answer: 216 `-:` 3

Problem: 2

Which of the following is equal to 12 - 5

a) 5-2

b) 17-10

c) 10-8

d) 76-66

Answer: 17-10

Monomial Math Term

Introduction of monomial math term:

Polynomial:

In mathematics, an expression which contains variables and constants is known as polynomial. Based on the number of terms in the polynomial it is divided into three types. They are monomials, binomial and trinomials. In this article we will see the binomials and monomials.

Monomials: A monomial expression contains one term in it.

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monomial math term


Monomial:

In mathematics, a term monomial is an algebraic expression which has a single term as an expression. The word ‘mono’ gives the meaning of only one.

For example, 888y is considered as a monomial term. A single integer digit is also considered as a monomial. Here the monomial term is derived as a variable along with constant number.

For example,12000

Limitation in the term monomial:

A monomial expression does not contain a negative sign and fractions in it.
Possible chances in monomials:

If the  monomial multiplied with other monomial and the result is also monomial.
If the monomial  multiplied with constant and the result is also monomial.
We can multiply the  monomial  with other types of polynomial such as binomials, trinomials.

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Examples for monomial math term:


Example:1

What is the monomial term among the following.

a ) 12-9

b ) z1/4

c ) 56x + 1

d ) 13 y

Solution:

Step 1: The first choice has a negative term in it, so it is also not a monomial term.

Step 2: The second choice has a fraction in it, so it is also not a monomial term.

Step 3: The third choice has two terms in it, so it is also not a monomial term.

Step 4: The fourth choice has variable with constant, so it is called as the monomial term.

Ans: 13y

Example 2:

What is the monomial term among the following.

a ) z +x

b ) 58y

c ) 56 - y

d ) 4 x + 3y + 2

Solution:

Step 1: The first choice has two terms in it, so it is also not a monomial term.

Step 2: The second choice has a one term in it, so it is a monomial term.

Step 3: The third choice has two terms in it, so it is also not a monomial term.

Step 4: The fourth choice has three terms in it, so it is also not a monomial term.

Ans: 58y