Probability

Probability:

            Probability is the study of random or nondeterministic events.
  e.g.  if we toss a fair coin, then it results either a head or tail; of course we cannot be sure of the outcome in advance.
              N=the number of tosses
              n=the number of heads of obtained,
            P(n)=1/2

Statistical Experiment:

       A random or statistical experiment is one in which
i) all possible outcomes of the experiment are known in advance.
ii) A performance of an experiment results in an outcome which is not known in advance.
iii) the experiment can be repeated under identical conditions.

Sample space:

    The sample space of an experiment is the set of all possible outcomes of the experiment.

   e.g. if we toss a coin twice,the sample space will be 

        S={hh,ht,th,tt}

Event:

    A subset of a sample space is called an event. An event is said to occur if an element of the event occurs

   e.g. E={hh,ht}

Prob. of an event:

       If the sample space S is finite, the probability of an event A denoted by P(A) is defined as 

                              P(A)=size of A/size of S

A S T C Rule:

A S T C rule:

    The abbreviation for 'all sin cos tan' rule in trigonometry is ASTC . The first letter of the first word in this phrase is 'A'. This may be taken to indicate that all trigonometric ratios in the first quadrant are positive . The first letter of the second word S indicates that sine and its reciprocal are positive in the 2nd quadrant. The first letter of the third word T indicates that tangent and its reciprocal is positive in the third quadrant. The first letter of the last word C indicates that cosine and its reciprocal are positive in the fourth quadrant.

                    

A S T C

                                                                                           

All Sin Cos Tan Rule:

1) When $\Theta$ lies in the first quadrant then all trigonometric ratios are positive.
2) When $\Theta$ lies in the second quadrant then sine and cosecant are positive other trigonometric ratios sin ,cosec, tan and cot are negative.
3) When $\Theta$ lies in the third quadrant then tangent and cotangent are positive other trigonometric ratios sin ,cosec, tan and cot are negative.
4) When $\Theta$ lies in the fourth quadrant then cosine and secant are positive other trigonometric ratios sin ,cosec, tan and cot are negative.

Types of triangles based on internal angle.

Types of triangles:

(Based on the internal angles)

  Based on the internal angles the triangles are of 3 types.
        1-Acute triangle
        1-Right triangle
        2-Obtuse triangle

1-Acute triangle:

         An acute triangle is a triangle with three acute angles. 

Acute triangle.

2-Right triangle:

          A right triangle or right-angled triangle is a triangle in which one angle is a right angle. The relation between the sides and angles of a right triangle is the basis for trigonometry. The side opposite the right angle is called the hypotenuse. The sides adjacent to the right angle are called legs.
           

         Right triangle

3- Obtuse triangle:

     An obtuse triangle is a triangle with one obtuse angle and two acute angles. Since a triangle's angles must sum to 180° in Euclidean geometry, no Euclidean triangle can have more than one obtuse angle.

Obtuse triangle

                   

Types of triangles based on side.

Types of triangles:

   Based on the sides of the triangle,the triangles are of 3 types.
               1-Equilateral 
               2-Isosceles
               3-Scalene
               

     1-Equilateral Triangle:
     

        The Equilateral triangle shown on the left has three congruent sides and three congruent angles.

Each angle is 60°.

     2-Isosceles Triangle:

               The Isosceles triangle shown on the left has two equal sides and two equal angles.

     3-Scalene Triangle:

           

       

               

The Scalene Triangle has no congruent sides. In other words, each side must have a different length.

                  

Pythagoras formula

Pythagoras formula:

 In Mathematics, the Pythagorean theorem, also known as Pythagoras' theorem, is a fundamental relation in Euclidean geometry among the three sides of a right triangle. It states that the area of the square whose side is the hypotenuse is equal to the sum of the areas of the squares on the other two sides.

Formula:

	=	side of right triangle
	=	side of right triangle
	=	hypotenuse

Triangle inequality

Triangle inequality:

 The triangle equality states that for any triangle, the sum of the lengths of any two sides must be greater than or equal to the length of the remaining sides.