Probability Simple Event

Probability is a method of expressing information or principle that an event will occur or has occurred. In mathematics the ideas has given a correct meaning in probability theory  is used widely in the areas of study as mathematics, statistics, science, and finance to depict conclusions about the possibility of potential events .An event is a group of possible outcomes.

A simple event in a probability is the event of a single outcome.

Simple Event

The simple event is as follows:

If we bring together a deck of 52 playing cards and no jokers, and pick a single card from the deck, then the sample space is a 52-element set, as each individual card is a possible outcome.

An event is any subset of the sample space, as well as any single-element set, the empty set and the sample space itself which is defined to have chance one.

Further events are correct subsets of the sample space that surround multiple elements. So, for example, potential events include:

Black and red at the same time without being a joker (0 elements),
The five of Diamond(1 element),
A Queen(4 elements),
A Face card(12 elements),
A Spade (13 elements),
A card(52 elements).

Seeing as all events are sets, they are generally written as sets (e.g. {1, 2, 3}), and symbolized graphically using Venn diagrams. Venn diagrams are mainly useful for symbolizing events because the probability of the event can be identified with the ratio of the area of the event and the area of the sample space.



Example:

The event that 1 roll a 7 with two dice is an event.

But the event that 1 roll snake-eyes (1 and 1) is a simple event.

A complementary event is just the opposite of an event.

The balancing event to rolling a 7 is the event that 1 do not roll a seven. It consists of all the outcomes not in the unique event.

Hypothesis Test Power

A statistical hypothesis test is a technique of making decisions using experimental information. In statistics, an effect is called statistically significant if it is doubtful to have happen by possibility. Critical examination of this category may be called tests of significance, and when such examination is available we may determine whether a next sample is or is not significantly dissimilar from the initial. Let us study about the topic hypothesis test power given below content with some example problems.


Example problems for hypothesis test power:

Example 1:

A sample of 900 members is establish to have a mean of 3.4 cm and standard deviation 2.61 cm. is the example in use from a great population of mean 3.25 cm and standard deviation 2.61 cm. If the population is usual and its mean is unknown, Get 95% self-confidence limits of exact mean.

Solution:

Given:

H0 : µ = 3.25
H1 : µ ? 3.25
L.O.S  a = 0.05

Test Statistic:

Z = (barx-mu)/(sigma)/(sqrt(n))

Given bar x = 3.4, µ=3.25, n=900, s=2.61

Z = (3.4- 3.25)/(2.61)/(sqrt(900))

= 1.724

Critical value:

At 5% level, the table  value of Z = 1.96

Conclusion:

Since |Z| <1 .96="" 5="" accepted="" at="" h0="" is="" level="" of="" p="" significance.="">
The sample is taken from population whose mean is 3.25 cm

Mean µ is unknown, 95% confidence limits of µ are = barx+-1.96 sigma/(sqrt(n))

=3.4+- 1.96*(2.61)/(sqrt(900))

= 3.4+- 0.17

=3.23,3.57

Example 2:-using hypothesis test power

A construct maintains that his artificial fishing line has a mean contravention strength of 8 kg and standard deviation 0.5 kg . Can we consider his state if a random sample of 50 lines yield a denote contravention power of 7.8kg?

Solution:

Given:

H0 : µ = 8
H1 : µ ? 8
L.O.S: a = 0.01

Test Stastistic

Z =(barx-mu)/((sigma)/(sqrt(n)))

Given:

barx = 7.8, mu = 8, n = 50, sigma = 0.5

Z = ((7.8)-8)/((0.5)/(sqrt(50)))

= - 2.828

|Z| = 2.828

Critical value:

At 1% level of significance the table value of Z = 2.58

Conclusion:

Since |Z|>2.58,H0 is rejected at 1% level.

The manufacture’s claim is not accepted.


Practice problem for hypothesis test power:

Problem 1: using hypothesis test power

On a test given 60 students at a huge number of several colleges, the mean position was 74.5 and S.D was 8.0. at one exacting college, where 200 students took test, the mean position was 75.9%. Discuss the meaning of this answer at 5% level.

Solution:

Z = 2.45, significant at 5% level.

Problem 2: using hypothesis test power

A random sample of 400 items is pinched from a normal population whose mean is 5 and whose variation is 4. if the sample mean is 4.45, can the sample be regard as truthfully random sample?

Solution:

Z= 5.5; H0 is rejected , No.

Standard Deviation

Standard Deviation is defined as "the measure of the variability". If the spread is more then the standard deviation will also be more. Standard deviation can also be stated as the "square root of the variance". Standard deviation has its application in computing the the annual returns of the banks

Arithmetic Mean:

A mathematical representation of the series of the number which is totalled as the sum of all the number in series and it is divided by the total number of values in the set.

Variability :

When the  standard deviation is raised to the power of two , the value which we get is the variance i.e Variance is equal to the square of the standard deviation. It is the measure of the degree of the spread among certain set of the values. Variance is also stated in the other way as follows"Variance is the measure of tendancy of individual values that vary from the arithmetic mean value

Formula to determine the Standard deviation :

[Formula of Standard deviation]

Where  ,     Σ = sum of

xi = Individual scores

m = arithmetic mean of all scores

n = Size of the number of scores

Variance :

Variance = σ2

Example

Example 1: To find the standard deviation of  2 , 4 , 6 , 8 , 10

Step-1 :Calculate the mean and the deviation

xi       m        xi-m         (xi-m)2

2        6        -4              16

4         6        -2              4

6        6          0              0

8        6          2              4

10       6         4             16

Step-2 :Find the sum of  (x-m)2

16 + 4 + 0 + 4 + 16 = 40

Step-3 :n = 5 , the total number of values , find n-1

5-1 = 4

Step-4 :Now find standard deviation

√40 / √4 = 3.162

3.162 is the standard deviation

To find the variance just square the standard deviation value

Step-5 : Variance = (3.162)2 = 9.998244

Example 2

Example 2 :To find the standard deviation of 3 , 6 , 9 , 12 , 15

Step-1 : Calculate the mean deviation

xi          m          xi-m           (xi-m)2

3           9           -6               36

6           9           -3                 9

9           9            0                 0

12          9            3                 9

15          9           6                36

Step-2 :

calculate the sum of (x-m)2

36+9+0+9+36 = 90

Step-3 :

n=5 ,the total number of values , find n-1

5-1 = 4

Step-4 :

Now let us find the standard deviation

√90/√4 = 4.743

The standard deviation is 4.743

Step-5 :

Variance = (4.743)2

= 22.496

Non Terminating Decimals

In this page we are going to discuss about Non terminating decimals.Below we will get introduction to Non terminating decimals. A non terminating decimal is one of the type of decimals which are divided into two types. They are,

Terminating Decimals
Non terminating Decimals

Non terminating Decimals is the continuous decimal number. It has been going on till the value infinitive. It is the very big value sometimes these non terminating decimals are rounded to its nearest values. But the Terminating decimal number is the fixed value of the decimal value.

Calculating Non Terminating Decimals

Below you can see the steps of calculating non terminating decimals:

Divide 5/3

Step 1: This is the type of an improper fraction. because here numerator takes the large value than the denominator.

Step 2: One time 3 is 3. So the remainder is 2 and the quotient value is 1

Step 3: Now 2 is not divided by 3

Step 4: So we can add zero to the remainder 2 and put a point the quotient value 1

Step 5: Now the value is 20. Again we can do the same procedure

Step 6: Now 6 times 3 is 18 again the remainder is 2 and the quotient value is 6

Step 7: Again we can add zero to 2 then the value is 20

Step 8: Again 6 times 3 is 18. Again the remainder is 2 and the quotient value is 6

Step 9: Now the decimal value is 1.66 …….its going on

Step 10: This decimal part 1.66……is known as the non terminating decimal.

Step 11: It can be rounded as 1.67

Step 12: Because it’s decimal part value is greater than or equal to 5. So the value of the last decimal value is consider as one and that value is added to the previous decimal value.

Step 13: Otherwise the last decimal part value is less than five. So the value of the last decimal value is simply leave. And the previous decimal value is considered as the last decimal digit value.

Pictorial Representation of this problem is shown below:

Non- terminating decimals

Practice problems

Here are some practice problems on non terminating Decimals:

Identify which of the fractions gives the non terminating decimals:

(4)/(2)
(2)/(3)
(6)/(3)

Answer:

Terminating decimals (4)/(2) =2 and (6)/(3) =2

Non terminating decimal (2)/(3) = 0.6666.....

vertices of a pyramid

A vertex of a polyhedron is the point of intersection of three or more faces of the structure.

A pyramid is a polyhedron with a polygonal base connected to a single point called the apex.

The number of vertices of a pyramid varies with the different types of pyramids.


How to determine the number of vertices

In a pyramid, the side faces (originating from the apex) meet at the base.

pyramid

So basically, for the square pyramid shown above, the four vertices of the square base become four vertices of the pyramid.

In addition, we have the apex, where the slant surfaces meet.

Hence, in a pyramid with a n-sided polygon as base, we have (n+1) vertices.

Some examples of vertices of a pyramid.

Now let us try to find the number of vertices of some common pyramids.

triangular pyramid

This is a triangular pyramid, i.e., the base is a triangle (3-sided polygon).

Hence, the number of vertices would be (3+1) = 4


rectangular pyramid

This is a rectangular pyramid, i.e., the base is a rectangle (4-sided polygon).

Hence, the number of vertices would be (4+1) = 5


pentagonal pyramid

This is a pentagonal pyramid, i.e., the base is a pentagon (5-sided polygon).

Hence, the number of vertices would be (5+1) = 6


Thus, it is quite simple to find the number of vertices of a pyramid given we know the type of polygon it has as its base. The square pyramid (with a square as its base) is the most common type of pyramid found in various monuments all over the world.

Construction of a Regular Pentagon

In this part we are going to see in detail about the introduction to construction of a regular pentagon. It’s easy to construct a regular polygon by simply using the compass and the ruler. And so is the construction of the regular pentagon. Most of the pentagons constructed in this method are done by inscribing it into the circle.

Construction of a regular pentagon Method 1:

Draw a circle of diameter AB centered at O.
Draw a line CD perpendicular to AB. Now mark the mid points of OA and OB as P and Q respectively. Draw circles with center as P and Q and radius OP and OQ respectively.
Draw a line from C to P. This line intersects the circle with center Q at T.
With C as center and CT as radius draw a circle. This circle intersects the main circle at E and F which are the successive pentagon vertices.
With length EF find the other vertices of the pentagon.

Since the pentagon has all its sides equal it will be easy to find the other sides if find any one of the vertices.

Construction of a regular pentagon Method 2:

• Draw a circle with center O and Diameter XX’.

• Then draw a line AQ perpendicular to the line XX’.

• Now Mark a point P such that it bisects the line OX’.


• With P as center and length AP draw an arc such that it cuts the line OX. Mark the point G.

• Then with AG as radius and A as center draw an arc such that it cuts the circle at B and C



• With AB or AC as radius find the points D and E.

• Join the points ABCDE to get pentagon.

Study Calculating Median

Study Median:

Median is the middle value of the element in their ascending order. If the size of the elements is even, then the average value of two middle numbers is the median. In the sequence of elements, divide equally the sequence of elements are on one side, half on the other, then the average value of middle elements is the median.

Study Calculating Median Formulas:

If ‘n’ is the total number of elements and ‘n’ is the ‘odd’, then the formula for finding the middle position after the arrangement of elements in ascending order is: (n+1)/2.

If ‘n’ is the total number of elements and ‘n’ is the ‘even’, then you have to find the two middle positions after the arrangement of elements in ascending order. Formulas for finding the two middle positions are: (n/2) and (n/2)+1. After finding the two middle elements, find the average that is the median.

Study Calculating median - Example Problems:

Study Calculating median Problem 1:

Calculate the median of 5, 4, 1, 8, and 7.

Solution:

Count the elements in the sequence has given.

n = 5(odd)

Arrange the elements in ascending order.

1, 4, 5, 7, 8

Formula for finding the middle position is (n+1)/2

(5+1)/2 = 6/2 = 3

Median = 3.
Study Calculating Median Problem 2:

Calculate the median of 8, 6, 1, 2, 3, 4, 7, and 5

Solution:

Count the elements in the sequence has given.

n = 8(even)

Arrange the numbers in ascending order.

1, 2, 3, 4, 5, 6, 7, 8

Find the two middle positions using n/2 and (n/2)+1

Positions are n/2= 8/2 = 4 and 5

4th and 5th position numbers are 4 and 5

The average is (4+5)/2 = 4.5

Median = 4.5

Study Calculating Median Problem 3:

Calculate the median of 2, 4, 6, 3, 1, 6, 3, 7, 4, and 6

Solution:

Count the elements in the sequence has given.

n = 10(even)

Arrange the numbers in ascending order.

1, 2, 3, 3, 4, 4, 6, 6, 6, 7

Calculate the two middle positions using n/2 and (n/2)+1

Positions are n/2= 10/2 = 5 and 6

4th and 5th position numbers are 4 and 4

The average is (4+4)/2 = 4

Median = 4

Study median practice Problems:

Study median practice problem 1:

Calculate the median of 4, 3, 1, 2, and 6.

Answer: 3
Study median practice Problem 2:

Calculate the median of 7, 9, 1, 4, 3, 6, 2, and 8

Answer: 5
Study median practice Problem 3:

Calculate the median of 5, 4, 6, 2, 3, 1, and 9

Answer: 5