Terms Related to Probability
The following are the terms related to describe the concept of probability:
Event
The outcomes of a random experiment is called its elementary event. When one coin is tossed, H and T is possible outcome. That is why Getting head is an elementary event. When two coins are tossed, the sample space of this experiment is {HH, HT, TH, TT} Events are HH, HT, TH and TT are the events of random experiment of tossing two coins.
Compound Events
When two or more than two elementary events occur with a random experiment, it s said to be Compound Event. When we throw a dice and getting odd number is a compound event.
Equally Likely Events
A given number of events are said to be equally likely, if none of them is expected to occur in preference to the others.
If you roll an unbiased dice then each number is equally likely to occur. If a dice is so formed that a particular face occurs most often then the dice is biased. In this case, the outcomes are not equally likely to happen.
Possible Outcomes
The total number of the events which are possible to occur is called possible outcomes.
When one dice is thrown, the possible outcomes are {1, 2, 3, 4, 5, 6}
When two dices are thrown , the possible outcomes are {(1,1), (1, 2), (1, 3), (1, 4), (1, 5), (1, 6), (2, 1), (2, 2), (2, 3), (2, 4), (2, 5), (2, 6), (3, 1), (3, 2), (3, 3), (3, 4), (3, 5), (3, 6), (4, 1), (4, 2), (4, 3), (4, 4), (4, 5), (4, 6), (5,1), (5, 2), (5, 3), (5, 4), (5, 5), (5, 6)}
Favorable Outcomes
The outcomes which satisfy the given condition of chance is called favorable outcomes. When an unbiased dice is thrown, the number obtained which is greater than 2 3, 4, 5, 6 are the favorable outcomes.
Probability of an Event
Let event is denoted by E the probability of occurrence ratio of event and sample space mathematically. Let the number of events is denoted by n (e) and number of sample space denoted by n(S) then\[P(E)=\frac{n(E)}{n(S)}\]
For examination you have selected 100 question for Preparation. In which 10 question match with exam paper. The probability\[=\frac{10}{100}=\frac{1}{10}\]
Sure Event
Sure event is an event whose probability is always one.
In a single toss of dice, what will be the probability of getting a number which is less then7?
Solution:
Sample space = {1, 2, 3, more... 
Introduction
The word 'probability" is one of the most commonly used word in our day to day life. Like probably today will raining, probably India will win the world cup etc. In mathematics, the concept of probability originated in the beginning of eighteenth century in problem of game of chance. Probability is a concept which numerically represents the degree of certainly of the event. Now a days it is widely used as the basic tools of statistics. Science and Engineering. 
Mode
Mode is the value that occurs the most of the time in a data or mode is a way of capturing important information about a random variable or a population in a single quantity. The mode is generally different from the mean and median.
Model Class
In a frequency distribution, the class having maximum frequency is called modal class.
Formula for Mode
Mode can be calculated by a formula, which is given below:
Mode \[={{x}_{k}}+h\left[ \frac{{{f}_{k}}-{{f}_{k-1}}}{2{{f}_{k}}-{{f}_{k-1}}-{{f}_{k+1}}} \right]\]
where \[{{x}_{k}}\]= lower limit of the modal class interval.
\[{{f}_{k}}\]= frequency of the modal class
\[{{f}_{k-1}}\]= frequency of the class preceding the modal class
\[{{f}_{k+1}}\]= frequency of the class succeeding the modal class
h = width of the class interval
Find the mode for the following frequency distribution:
| Class Interval | Frequency | Class Interval | Frequency | ||||||||||||||||||||||
| 0 - 10 | 5 | 40 - 50 | 28 | ||||||||||||||||||||||
| 10 -20 | 8 | 50 - 60 | 20 | ||||||||||||||||||||||
| 20 - 30 | 7 | 60 - 70 | 10 | ||||||||||||||||||||||
more...
 Median
Median of a data is the value of the variable which divides it into two equal parts. It means it is the value of variable so that the number of observation above it is equal to the number of observation below it. Suppose\[{{x}_{1}},{{x}_{2}}.....{{x}_{n}}\]are n observation in ascending or descending order. The median of the above observation is:
(i) n is odd then median is the value of\[{{\left( \frac{n+1}{2} \right)}^{th}}\] observation.
(ii) n is even then median is the value of arithmetic mean of \[{{\left( \frac{n}{2} \right)}^{th}}\]and \[{{\left( \frac{n}{2}+1 \right)}^{th}}\] observation i.e
\[\text{Mean}=\frac{{{\left( \frac{n}{2} \right)}^{th}}\text{observation}+{{\left( \frac{n}{2}+1 \right)}^{th}}\text{observation}}{2}\]
Method for finding the median for grouped data. Step for finding the median
Step 1: Forgiven frequency distribution, prepare the commutative frequency table and obtain\[N=\sum {{f}_{i}}\].
Step 2: Find (N/2).
Step 3: Look at the cumulative frequency Just greater than (N/2) and find the corresponding class, known as median class.
Step 4: Then by using median formula, calculate median, which is given below:
Median \[=l+\left[ h\times \frac{\frac{N}{2}-C}{f} \right]\]
where I = lower limit of median class, h = width of median class,
f = frequency of median class
c =cumulative frequency of the class for preceding the median class
\[N=\sum fi\]
Find the median class of daily wages from the following frequency distribution
|
Mean
It is also known as arithmetic mean of a given observation is equal to ratio of sum of observation and total number of observation, i.e.
\[\text{Mean }=\frac{\text{Sum of total observation}}{\text{Total Number of observation}}\]
If \[{{x}_{1}},{{x}_{2}},.....{{x}_{n}}\] are n observation then its mean
\[\overline{\text{X}}=\frac{{{x}_{1}}+{{x}_{2}}+.....+{{x}_{n}}}{n}=\frac{\sum {{x}_{i}}}{n}\]
The arithmetic mean of grouped data calculated by the following methods:
(i) Direct method
(ii) Assumed mean method
(iii) Step - Deviation method
Direct Method
In this method, suppose \[{{x}_{1}},{{x}_{2}},.....,{{x}_{n}}\] are the observation having frequency\[{{f}_{1}},{{f}_{2}},.....,{{f}_{n}}\] respectively then
\[\overline{\text{X}}=\frac{{{x}_{1}}{{f}_{1}}+{{x}_{2}}{{f}_{2}}+.....,+{{x}_{n}}{{f}_{n}}}{{{f}_{1}}+{{f}_{2}}+.....{{f}_{n}}}=\frac{\sum {{x}_{i}}{{f}_{i}}}{\sum {{f}_{i}}}\]
Direct method for calculating mean depend on the following steps:
Step 1: Find the class mark (which is discussed in previous class) for each class.
Step 2: Calculate product of frequency and class mark for each class interval.
Step 3: Then calculate mean by using formula \[\left( \frac{\sum fixi}{\sum fi} \right)\].
Find the mean of the following data:
Introduction
Previously we have studied about the representation of data. Data handling in an art. It depends upon expertise or it. In this chapter we will emphasis on measure of central tendencies of data.
Introduction
This is the one of the fundamental concept of "Trigonometry". Trigonometry is the branch of mathematics which deals with the measurement of sides and angles of a triangle. In modern days, its scope has been extended and it also includes the study of polygons and circles.
Suppose when a point P put anywhere in the above line, it will be converted into rays. These rays are responsible for the formation of an angle means "An angle is a geometrical figure made by two rays having common end point (called vertex)" these rays are called sides or arms of an angle. An angle is represented by S a symbol "<". The way to represent it put vertex in the middle.
\[\angle \text{RPO},\angle \text{SPQ},\angle \text{RPS}\] are the angles with vertex P.
The inclination of one ray to other represented by a number is called measurement of an angle. One of the most important aspect of geometry is that all the angles are between \[0{}^\circ \]and\[\text{36}0{}^\circ \] and there is no meaning of negative angle. 
Concept of Angles in Trigonometry
In case of trigonometry, angles may be positive or negative and of any magnitude.
Positive Angles
When we measure an angle in anti-clock wise direction it is always positive.
\[\text{PQ = R}\]
We know that \[\text{ }\!\!\pi\!\!\text{ }\] is the ratio of circumference and diameter.
\[\frac{\text{C}}{\text{D}}\,\,\text{=}\,\text{ }\!\!\pi\!\!\text{ }\]
Relation between radian and degree, \[\pi \]radian\[={{180}^{o}}\]
It is also written as
\[{{\text{ }\!\!\pi\!\!\text{ }}^{\text{C}}}\,\text{=}\,\text{180 }\!\!{}^\circ\!\!\text{ }\]
Relation between Angle, Length of an Arc and Radius of Circle
We know that the angle formed at the centre is proportional to the length of arc which subtend it.
Means \[\theta \,\,\,\,\alpha \,\,\,\,\,\,l\]
(Here,\[\theta \]represents angle and\[l\]represents the length of arc)
By the definition of radian.
Angle subtended by an arc of length r at the centre of circle = 1 radian.
\[\therefore \] Angle subtended by an arc of length 1 at the centre of circle \[=\frac{1}{\pi }\] radian.
\[\therefore \] Angle subtended by an arc of length (. at the centre of circle \[=\frac{\ell }{r}\]
\[\therefore \] \[\mathbf{\theta = }\frac{\mathbf{I}}{\mathbf{r}}\]
Relation among Different Units of Measurement of an Angle
1. \[{{1}^{o}}=\frac{\pi }{180}\]radian
2. 1 radian \[={{\left( \frac{180}{\pi } \right)}^{0}}\]
3. \[{{\left( \frac{10}{\pi } \right)}^{g}}={{\left( \frac{\pi }{{{180}^{o}}} \right)}^{c}}=1\] degree
It is convention that angles are always measured either in radian or in degree.
Conversion of Some Common Angle in Degree into Radian
Area of Plane Geometrical Figures
Area of Plane Figures
The area of a plane figure is the measurement of the surface enclosed by its boundry. In this chapter we will study about different plane figures with its area.
Area of Triangle
Area of \[\Delta \text{ABC}=\frac{1}{2}\text{BC}\times \text{AD}\] square unit
Area of right triangle \[=\frac{1}{2}\times \text{(perpendicular)}\times \text{Base}\] \[=\frac{1}{2}\times AB\times \text{BC}\]
Area of Triangle by Heron's Formula
Let a, b, c be the length of sides of a triangle
then area \[=\sqrt{s(s-a)(s-b)(s-c)}\]sq. unit where \[s=-\frac{1}{2}(a+b+c)\]
Area of Equilateral Triangle
Area \[=\frac{\sqrt{3}}{4}{{(side)}^{2}}=\frac{\sqrt{3}}{4}{{a}^{2}}\] Area of isosceles triangle
\[=\frac{1}{2}\times BC\times AD=\frac{1}{4}b\sqrt{4{{a}^{2}}-{{b}^{2}}}\]
Find the area of the triangle whose base = 25 cm and height = 10.8 cm.
(a) \[\text{125c}{{\text{m}}^{\text{3}}}\]
(b) \[\text{135c}{{\text{m}}^{\text{2}}}\]
(c) \[\text{124c}{{\text{m}}^{\text{2}}}\]
(d) \[\text{199}\,\text{c}{{\text{m}}^{\text{2}}}\]
(e) None of these
Answer: (b)
Explanation
Area of the given triangle
\[\text{=}\frac{1}{2}\times \text{base}\times \text{height}=\left( \frac{1}{2}\times \text{25}\times \text{1}0.\text{8} \right)\text{c}{{\text{m}}^{\text{2}}}=\text{135 c}{{\text{m}}^{\text{2}}}\] 
Circle
We know that circle is locus of a point which moves in plane in such way that its distance from a fixed point is always constant. In this section we will emphasis on sectors, segment etc.
Sector of a Circle
The region enclosed by an arc of a circle and its two bounding radii, called sector of the circle.
In the above given figure. OABC is a sector of the circle with centre 0. Here AC is the minor arc therefore, OABC is the minor sector and other plane sector is the major sector. Segment A segment of a circle is the region bounded by an arc and a chord. The segment containing minor arc is called a minor segment. While the segment containing the major arc is the major segment. The centre of the circle lies in the major segment.
Arc
A continuous piece of a circle is called an arc of the circle. In the figure given y below AB is an arc of a circle with centre O, denoted by \[\overset\frown{AB}\]. In the figure given I below \[\overset\frown{ACB}\] is the minor arc and \[\overset\frown{ADB}\] is the major arc.
Central Angle
An angle subtended by an arc at the centre of a circle is called central angle. In the figure given below of a circle with center 0 central angle of \[\overset\frown{AB}=\angle AOB=\theta \]. If \[\theta <180{}^\circ \] then the arc AB is called the minor arc and the arc BA is called the major arc.
