Statistical Probability

Let us study about Statistical Probability,
One of the most familiar uses of statistics is to determine the chance of some occurrence. For instance, what are the chances that it will rain tomorrow or that the Boston Red Sox will win a World Series? These kinds of probabilities, although interesting, are not the variety under discussion here. Rather, we are examining the probability in statistics that deals with classic theory and frequency theory—events that can be repeated over and over again, independently, and under the same conditions.

Coin tossing and card drawing are two such examples. A fair coin (one that is not weighted or fixed) has an equal chance of landing heads as landing tails. A typical deck of cards has 52 different cards—13 of each suit (hearts, clubs, diamonds, and spades)—and each card or suit has an equal chance of being drawn. This kind of event forms the basis of your understanding of probability and enables you to find solutions to everyday problems that seem far removed from coin tossing or card drawing.
Hope the above explanation was useful.

Law of Cosines

Let us study the law of Cosines,
If α, β, and γ are the angles of any (right, acute, or obtuse) triangle, and a, b, and c are the lengths of the three sides opposite α, β, and γ, respectively, then


These three formulas are called the Law of Cosines. Each follows from the distance formula and is illustrated in Figure 1 .

Figure 1

Reference triangle for Law of Cosines.

From the figure,

Thus the coordinates of A are Remember, all three forms of the Law of Cosines are true even if γ is acute. Using the distance formula, In the preceding formula, if γ is 90°, then the cos 90° = 0, yielding the Pythagoras theorem for right triangles. If the orientation of the triangle is changed to have A or B at the origin, then the other two versions of the Law of Cosines can be obtained. Two specific cases are of particular importance. First, use the Law of Cosines to solve a triangle if the length of the three sides is known. Hope the above explanation was helpful.

square root of complex number

Let us learn about square root,

A divisor of a quantity that when squared gives the quantity. For example, the square roots of 25 are 5 and -5 because 5 × 5 = 25 and (-5) × (-5) = 25.

A number that, when squared, yields a given number. For example, since 5 × 5 = 25, the square root of 25 (written 25) is 5.

"Roots" (or "radicals") are the "opposite" operation of applying exponents; you can "undo" a power with a radical, and a radical can "undo" a power. For instance, if you square 2, you get 4, and if you "take the square root of 4", you get 2; if you square 3, you get 9, and if you "take the square root of 9", you get 3:

Hope the above explanation helped you.

distance formula in coordinate geometry

Let us study about distance formula in coordinate geometry,

In Figure, A is (2, 2), B is (5, 2), and C is (5, 6)

Finding the distance from A to C.

To find AB or BC, only simple subtracting is necessary.

AB = 5 − 2 and BC = 6 − 2 AB = 3 BC = 4

To find AC, though, simply subtracting is not sufficient. Triangle ABC is a right triangle with AC the hypotenuse. Therefore, by the Pythagorean Theorem,

If A is represented by the ordered pair ( x₁, y₁) and C is represented by the ordered pair ( x₂, y₂), then AB = ( x₂ − x₁) and BC = ( y₂ − y₁).

Then

This is stated as a theorem.

Theorem: If the coordinates of two points are ( x₁, y₁) and ( x₂, y₂), then the distance, d, between the two points is given by the following formula (Distance Formula).

Example 1: Use the Distance Formula to find the distance between the points with coordinates (−3, 4) and (5, 2).

Example 2: A triangle has vertices A(12,5), B(5,3), and C(12, 1). Show that the triangle is isosceles.

By the Distance Formula,

Because AB = BC, triangle ABC is isosceles.

Hope the above explanation helped you.

Circle and line Intersectionhttp://www.blogger.com/img/blank.gif

Let us learn about circle and line intersection,
So, let us consider a circle and a line PQ. There can be three possibilities given
in Fig. below:
In Fig. (i), the line PQ and the circle have no common point. In this case,
PQ is called a non-intersecting line with respect to the circle. In Fig.(ii), there
are two common points A and B that the line PQ and the circle have. In this case, we
call the line PQ a secant of the circle. In Fig.(iii), there is only one point A which
is common to the line PQ and the circle. In this case, the line is called a tangent to the
circle.