Types of events

Types of events in probability and statistics:
There are various types of events in probability and statistics:

1. Simple event: If the event consists of less than 2 elements from the sample space then it is called a simple event or elementary event.

If sample space is S = 2, 3, 4, 5, 6.
And if event is E = 3, then it is a simple or elementary event.

1. Compound event: It comes under the types of events in probability and statistics as if the event consists of more than one element or sample points from the sample space then it is called a compound event.

If sample space of throwing a dice is, S = 1, 2, 3, 4, 5, 6.
And if event is, E = 1, 3, 5, then it is known as a compound event.

1. Sure event or certain event: It comes under the types of events in probability and statistics (for more visit this)as if the sample space is, S = 1, 2, 3, 4, 5, 6 and the event is also same as that of sample space, i.e., E= 1, 2, 3, 4, 5, 6 which is also the subset of S and it occurs whenever an experiment is performed. Therefore, the event E is called as a sure event or certain event.

1. Mutually exclusive events: It comes under the types of events in probability and statistics as  if we consider a random experiment of throwing a dice and sample space is, S=1, 2, 3, 4, 5, 6. Two events associated with the above experiment are called mutually exclusive if both the events cannot occur at the same time or in the same trial. Like A=1, 3, 5 and B=2, 4, 6 are mutually exclusive events. If the events are A = 1, 2, 4 and B= 2, 5, 6 are not mutually exclusive as 2 is favorable to both the events.

1. Exhaustive Events: It comes under the types of events in probability and statistics for any random experiment, S, E1, E2, E3,….En are the events which are the subsets of S.

E1, E2, E3,…..En are consider as exhaustive events if the union of all of them is equal to S, i.e.,
E1 U E2 U E3 U ….U En = S
The above equation also proves true if the set of events E1, E2,…En are mutually exclusive and exhaustive events.

1. Null event: This event is always added in the sample space as a sample point called φ(fi).  This event is the subset of S and is called as null event or impossible event.

In upcoming posts we will discuss about Permutations and combinations and Intersection of a plane with 3-d figures. Visit our website for information on Measures of Central Tendency

Probability and Statistics

Probability And Statistics : when we study in secondary school education Andhra Pradesh about the approaches of studying the theory of  Probability they are  Statistics approach and the classical approach of finding probability.  In statistics approach we find the probability of occurrence of an event by observing the repeated experiments and their frequency. Here we find the ratio of  the frequency of the event to be observes to the total frequency of the event which has been observed. Let us consider an event of throwing a coin. The possible outcomes of this event are Head and Tail, called as H and T.

The probability of getting a head on a throw of a coin will be the ratio of frequency of getting a Head to the total outcomes. Here we have total outcomes = 2 and the frequency of getting a head in total outcomes is 1. SO the probability of getting a head = 1/2. Similarly we can find the probability of getting a tail is also 1/2. This method of finding the probability applies to  the cards, dice, coins and other many simple games of chance. When we are diverted from these games of chance and consider the situations which are likely to occur like probability of people who will die before age of 60, probability of getting a rain this week. In such cases it is not possible to do simple enumeration of cases which are likely to occur.

In upcoming posts we will discuss about Types of events and Learn Pythagorean Theorem. Visit our website for information on Stem and Leaf Plot

Math Blog on Grade XI

Triangles:
Out of many geometrical structures, triangle is one of the basic shapes of geometry. Triangles are polygons having 3 sides or edges, which are nothing but line segments and 3 angles or vertices, which may be of same measures sometimes (for quick solution also try area of equilateral triangle calculator). The sides are termed as A, B, C and triangle is represented as Δ ABC.
As all grade XI board of intermediate education Andhra Pradesh students are aware, triangles are classified according to the lengths of the three sides:
1.      Isosceles triangle: Here, two sides are same in length and two angles which are opposite to the sides of same length are same in measure.

2.      Equilateral triangle: All sides are equal in length and all three angles are same measuring 60 ⁰ each (as total interior angle in a triangle is 180⁰) in case of equilateral triangles.


3.      Scalene triangle:All sides are different in length and all three angles are different in measure in case of scalene triangles.

Triangles are also classified according to the angles of three sides (for more on triangles visit this):
1.      Right angle triangle:These triangles has one of its interior angle measuring 90⁰ . In case of right triangles, special names are given to each side of the triangle, with the side which is opposite to the right angle being termed as  hypotenuse, which is the longest side of the right triangle and the other two sides being known as the legs.

2.      Acute  angle triangle: All three interior angles are less than 90⁰ in case of acute angle triangles.

 


3.      Obtuse angle triangle:These triangles have one angle which is more than 90⁰ .

In grade XI you will also be studying one more topic, i.e., special right triangles, which are nothing but right triangles having some specified features which makes calculation easy. There are two types in which special right triangle are being classified: angle based and side based.
In angle based special right triangle, the angles are divided such that the right angle (largest) is equal to sum of the other two angles. There are two types of  special right triangle which are angle based:
1.      45-45-90 angle:  As you can judge by name, it has 2 angles measuring 45⁰ and one 90⁰ and you can get such triangle by cutting a square diagonally. Here the angles are in the ratio 1 : 1 : 2 and by using Pythagoras theorem(which says - in a right angle triangle, the square of the length of the hypotenuse is equal to the sum of the square of the lengths of the two legs.  The theorem is represented by the equation: a² + b² = c², where c represents the hypotenuse, longest side) , the sides are in the ratio of
1: 1 : √2, which means that two sides are equal in length also.
 Hence such kind of triangles have characteristic of both the isosceles and the right angle triangles.


2.      30-60-90 angle: This kind of special right triangles are very common and has one 30⁰ ,one 60⁰ and one 90⁰ angle.  There is one theorem which applies to these triangles, stated as,

In a 30-60-90 triangle, the  length of hypotenuse is two times the length of leg opposite to the 30^o angle. And the length of the other leg is calculated as SQRT (3) times the leg opposite to the 30^o angle.




Now finally, in side-based special right triangles, the lengths of the sides form ratios of whole number, such as 3 : 4 : 5 ,11 : 12 : 13 etc.

In upcoming posts we will discuss about Probability and Statistics and Congruence. Visit our website for information on Calculating Standard Deviation

Permutations and combinations

Hi friends, we are discussing the topic Permutations and combinations of ssc board Andhra Pradesh in this blog. The combinations and permutations are part of algebra which is part of Discrete math.  In combination the order of things is not important, example: the fruit salad is combination of papaya, mango, grapes, apple we do not care what order of fruits is in.
The combination is two types: 1) repetition 2) not repetition.
Allowed the Repetition: in your pocket the number of coins can be repeated (2, 2, 2, 5, 5, 10, 10)
Repetition is not allowed: in the lottery numbers repetition is not allowed (5, 67, 45, 96, 24, 54)
Example of the combination: when we are choosing balls the possibilities of selection can be 1, 2, 3 balls.
Order does matter: 123, 132, 231, 321, 312, 213. The formula of combination is ^aC_b
            a!               (! this symbol is used for the factorial operation)
^aC_b =---------
       b!(a-b)!      
Here, a is the number of things from which we have to choose b number of things.
The formula is applied on example where we have to choose 3 out of 5 different balls:
   5!                5!             120
 ----------- = ---------  =  10 ways of combination.
  3!(5-3)!     3!*2!            12
Permutation is an order of combination in which the order does not change (visit for more information on permutation). The lock is the best example of permutation; the permutation is totally different to the combination. Permutations are basically of two types:

1.      Repetition is compulsory: the code of the lock is: 555. The repetition is allowed in this case and order does not matter.

Simple formula for repetition: a^b
Here a*a*a*.........(b times)=a^b

2.      No repetition allowed: In this we have to reduce number of available choices for each time

Example: in the pool game number of balls is fixed. One ball is used one time and no repetition is allowed for using the same ball. The formula of no repetition permutation:
            a!         (!- this symbol is used for the factorial operation)
 ^aP_b=-------
         (a-b)!
An Example of this permutation can be when we have to pick 2 out of 5 balls in all possible ways:
5!           5*4*3*2*1      120
---- – = ---------- ---- = 20 ways
(5-2)!        3*2*1          6


In upcoming posts we will discuss about Math Blog on Grade XI and Properties of quadrilaterals. Visit our website for information on How to Calculate Standard Deviation

Correlation and causation

Correlation and causation for Grade XI of AP secondary school education board:
To understand Correlation and causation let’s have a glance at the following example related to Correlation and causation.
• When ice cream sales increase, the rate of drowning deaths increases too in accordance with increase in the sales of ice creams.
Conclusion: ice cream consumption causes drowning deaths.
Explanation: ice cream sales increases during the summer time not in cold and in summer time people are also involved in the water related activities like swimming etc. Thus drowning deaths happen due to more exposure to water based activities not because of ice cream. Thus the conclusion is false.
Linear Correlation versus causation:
There is a great confusion between Correlation and causation. It can be specified easily. An event can cause another event like smoking causes lung cancer and it can be correlated with another factor like smoking is also co related with alcoholism. If the occurrence of the event causes another event then definitely they are co related but it’s not necessary that if occurrence of two things is together one will always cause the other. Basically it’s difficult to establish causality between two correlated events. Like in the above example smoking not only correlates with lung cancer but it causes lung cancer.
Most effective method of defining causality is the controlled study in which two groups of people are given two different sets of experiences (remind that these groups of people chosen are same in most of the ways) like one group decides to watch opera soaps and other watches video shows and then the outcome of these groups is compared. If outcomes of both the groups are different then it is believed that the different experiences of the groups would have caused different outcomes.

In upcoming posts we will discuss about Permutations and combinations and Angles of triangles and polygons. Visit our website for information on What is Standard Deviation

Statistical experiments

Statistical Experiments for Grade XI from Andhra Pradesh board syllabus:
Statistics (click this for more information on statistics)is the process of analyzing of data. It includes planning, analyzing, designing and at the end recommending the errors. Statistics is process of outcomes (also try descriptive statistics examples). Statistical Experiments have three things common:
• The experiment that is performed can have more than one likely outcome.
• The probable outcomes can be specified in advance as they are the events linked to probability factor.
• As said above, outcomes are the probability factors so each outcome of experiments depends on the chance.
For example- a coin toss; it has all the characteristics that are specified above. In a coin toss, there are more than one possible outcome i.e. (head, tail) and they can be specified in advance. The third point is the probability factor. As outcomes are uncertain, so their possibility is an element of chance.
It can be understood by another example.
Suppose we are rolling a die. Now first it should be confirmed that is it a Statistical Experiment. Yes. It is. As in the above example there is more than one possible outcome i.e. 1, 2, 3, 4, 5, and 6 and each outcome can be specified in advance and the possibility of the outcome is the matter of chance. The sample space of a single dice is 1, 2, 3, 4, 5, and 6. The numbers 1 to 6 are events in the sample space.
Let's have another example of Statistical experiments of more than one variable.
If we are rolling a single die two times then both these events are independent because the roll of a die at a time would not affect another roll of that die. This means that independent events are those events in which the occurrence of one event has no effect on the occurrence of another event.

In upcoming posts we will discuss about Correlation and causation and parallel lines cut by a transversal. Visit our website for information on Standard Deviation Formula

Methods of data representation

There are various methods of data representation like arrangement of data in tabulated form or in form of graphs or charts or maps or diagrams .These data representation methods described in grade XI of Andhra Pradesh sample papers are dependent on the nature of the data or nature of the analysis that is based on the data or others equal to it .We can understand each information that are having the same identical structure or having the same identification number. So anyone can easily find the data or value according to these specifications. We can take an example of GIS data that is used to represent the real objects as roads, trees, houses etc.
So data representation is a technique of capturing the raw data change them into the useful information and then represent them into the pictorial form or in tabular form. There are several methods like image processing , Global positioning system all are based on the various type of data representation .The main motive behind presenting the data is to understand the available data and deriving the meaning and also extract the useful in conclusion .The representation of data can be done in many ways like :
Statistical tables            
or by rank order
or by frequency order .
These forms are useful in statistical analysis of the data .Initially all the data collect in scattered form but later it organize and change into the numerical facts. (Improve your skills by playing data analysis worksheets)
Statistical tables: Data will arrange into the form of rows and columns. These tables are used to collect the raw data and also the percentage or mean or median and so on.

Rank Order: Data is present in ascending and descending order .On the basis of rank or position order will be exhibit .as an example
S.NO. Scores S.NO. Scores
1 20 3 18
2 19 4 14

Frequency Distribution: Sometimes Rank order not much useful to summarize the series of raw data and also not describe about the frequency so in that case frequency table is used and range can also be described in it.


In upcoming posts we will discuss about Statistical experiments and triangle inequality theorem. Visit our website for information on z score chart

Measures of central tendency in Grade XI

In this section we are going to discuss the topic Central tendency and how to calculate measures of central tendency. Central tendency for grade XI  of Andhra Pradesh board textbooks means calculation of the average data or number among the data set or large amount of data. Measuring central tendency is a process which has basically three types as Mean, Median and Mode. You can improve your math skills by playing measures of central tendency worksheets
First we discuss about the Mode that is the value that occurs many times in a data set. It can also have the frequency of the repeated numbers as an example a group of data set have the ten numbers as 92 , 100 , 103 , 98 , 100 , 95 , 100 , 99 ,100 ,92 then 100 is the number which occurs 4 times in the data set so 100 is the mode of this data set .
Another measure of central tendency is median that is defined as the value which is occurs in the middle of ordered data set. We also describe it by an example as 32 , 78 , 45 , 36 , 90 , 89 , 35 , 56 then when we arrange it in order as 32 , 35 , 36 ,44 , 56 , 78 , 89 , 90 then 45 and 56 is the middle values of ordered data set then the median of it is ( 44 + 56 ) / 2 = 100 / 2 = 50 .
There is another way of measuring central tendency (for more visit here) known as mean that is normally an arithmetic mean which is equal to the sum of the value s of data set which is divided by total number of values. It is often known as average. We can also understand it by an example as a data set of values as 4 , 6 , 8 , 3 , 5, 10have the mean value ( 4 + 6 + 8 + 3 + 5+ 10)/6= 36 / 6 =6 where total number of values are 6 and their total is 36 and when divide 36 by 6 then coming value is known as mean.

In upcoming posts we will discuss about Methods of data representation and Triangle congruence relationships. Visit our website for information on Z score Table

Measures of dispersions in Grade XI

Measure of dispersions (for more visit here) in the simplest form is the difference between the maximum and the minimum values which is defined in terms of range. We can write it as
Range = Maximum value – minimum value
If a variable x has three different values 39, 40, 41 then Range is 41 – 39 = 2. In another example if x has values 18, 40, 60 than range is 60 – 18 = 42. In first example x has small dispersion and in another example x has large dispersion. So it is the simplest form of dispersion.
But sometimes it is affected by the outliers and that time it will be a false measure of dispersion.
Another type of measuring statistics dispersion is IQR that is inter-quartile range that is defined as IQR = Q3-Q1
IQR is come in existence to overcome the effect of outliers on the range. IQR is the process that eliminates the outlier from the data in the way that it removes the lowest 25% of the values in ordered and also removing the 25% of the highest ordered values and then range of the remaining data is the Q3 – Q1.
One of the measures of dispersion is Standard Deviation. It is used more than any other dispersion method. It is described as it is the deviation of every value of the actual data from the mean of all the data. For avoiding the zero as sum, square the deviation. Now the question is How to Calculate Standard Deviation? For standard deviation first we calculate the variance and it is denoted by sigma ( σ² ) and the expression is
σ² = ∑ (X – μ )² / N and Standard deviation ( σ ) is σ = ∑ [(X - μ)² / N ]^1/2
Where μ is mean and σ² is variance.
Sometimes we want to compare the two sets of data, that time we use the relative measure of dispersion and we cannot compare the two sets of data until their units of measurement are not same. So by using the relative measure of dispersion we compare it easily as by calculating the coefficient of variation (CV) and it is calculated as CV= S / X * 100. Here s is standard deviation and x is mean of data set. So above are some measures of dispersions that are defined for grade XI.

In upcoming posts we will discuss about Measures of central tendency in Grade XI and Basic constructions. Visit our website for information on CBSE 11th syllabus

Math Blog on Types of events

The types of events in probability are independent, mutually exclusive or conditional (solve more Conditional Probability Problems here). An event means set of one or more outcomes.
Let us take an example. We throw a die. E1 is an event to get a perfect square number. Then E1 = 2, 4. Suppose that 3 comes up on the upper face, then this means that the event E1 has not occurred. Event occurs only when 2 or 4 appears on the upper face. Therefore, if an outcome satisfies the conditions, then we say that the event has occurred. If z is the outcome and E1 is the event of a sample space S’ and then event E1 has occurred if z belongs to E1.
In mutually exclusive two events associated with a random experiment, if both the events cannot occur together in the same trial. In the above experiment of throwing a die, the events A = 2, 4 and B = 1, 5, 6 are mutually exclusive events and in the same experiment, the events A = 2, 3, 4, and C = 2, 4, 5, 6 are not mutually exclusive because 2 and 4 appears on both events A and C. From this the definition of mutually exclusive events can extended to more than two events. If the happening of one rules out of these then we can say that more than two events are mutually exclusive. An events A = 1, 4, B = 2 and C= 5, are mutually exclusive with the experiment of throwing a single die.
Let A and B are two events, then A or B or (A ∪ B) represented the event of the occurrence of at least one of the events A or B. A and B or (A ∩ B) shows the event of the occurrence of both events A and B. If A and B happen to be mutually exclusive then events can be written as P (A ∩B) = 0. This is the type of events in probability for Grade XI.

In upcoming posts we will discuss about Measures of dispersions in Grade XI and circles. Visit our website for information on CBSE previous year question papers class 12