Probability Model

The probability model of an event is defined as a number between one and zero. In reality however there is practically nothing that has a probability of 1 or 0. The theoretical probability of an event E is the fraction of times we expect E to occur if we show again the same experiment over and over. The expected probability approaches the theoretical probability as the number of trials gets bigger and bigger. Thus, If E consists of a single outcome s, we refer to P (E) as the probability of the outcome s, and write P(s) for P (E).


Examples on Probability Model


Following are some examples on Probability Model.

Example 1:

A jar contains 5 white, 6 black, 4 green and 8 yellow marbles. If a particular marble is selected at random from the jar, what is the probability of choose a white marble? a green marble? a black marble? a yellow marble?

Solution:

Probability = number of way to choose white/total number of the marble = 5/23

Probabilty = number of way to choose green/total number of the marble = 4/23

Probability =number of way to choose black/total number of the marble = 6/23

Probability =number of way to choose yellow/total number of the marble = 8/23

Example 2:

A bag contains 6 orange, 4 red, 7 blue and 5 yellow balls. If a single ball is selected at random from the bag, what is the probability of choose a orange ball? a red ball? a blue ball? a yellow ball?

Solution:

Probability (orange) =number of way to choose orange/total number of the ball = 6/23

Probability (red) =number of way to choose red/total number of the ball = 4/23

Probability (blue) =number of way to choose blue/total number of the ball = 7/23

Probability (yellow) =number of way to choose yellow/total number of the ball = 8/23

Multiplication

Multiplications is fundamental operation on whole numbers. The fundamental operations on whole numbers are addition, subtraction, multiplications, and division. The time is the process of repeated addition of a number. Times or multiply denoted by the symbol × or *. Elementary school students use the symbol × for multiplications (multiply) or times. Leibniz uses the cap symbol for multiplications or times. This symbol is used to indicate intersection in set.
Multiplications

Multiplications indicates by placing the quantities to be multiplied side by side (just a position). The important of multiplications process is to place the digits value of the factors in the proper columns. That is, units number must be placed in the units column, tens in tens column, and hundreds in hundreds column. Notice that it is not essential to write the zero in the case of 15 tens (150) since the 1 and 5 are written in the proper columns.

Example 1:

Multiply 750 × 99

Solution:

750 × 99 means 99 times 750

or 100 times 750 – one time 750

or 75000 – 750

or 74250

Therefore 750 × 99 = 74250

Rules of linear equation

A linear equation is a statement of equality which contains an unknown quantity or variable. Any value of the variable which makes the statement true is called the solution or root of the equation. There are certain rules in this equation to solve the problems. These rules are applied in such a way that one side of the equation may have only unknown quantity.Now let us study about linear learning.

Rules for learning linear equations:

By learning linear equation of one variable the following rules are used which do not change the equation. They are as follows,

• The number which can be added on both the sides of an equation.

• Subtract the number from both sides of an equation.

• The number which can be multiplied on both sides to simplify the equation

• The number can be divided in that equation on either sides

Inverse Variation

The inverse variation is the product of two variables equals to a constant and the product is not equal to zero. Inverse variation is in the form of y =k/x. xy = k. Inverse variation in which value of one variable increases while the value of the other variable decreases in value is known as an inverse variation. For example think a trip of 240 miles. Rate(mph)=>20,30,40,60,80,120 and time(h)=>12,8,6,4,3,2.The numbers can be explained. As the rate of speed increase, the numbers of hour require decrease. As the rate of speed decrease, the number of hour requires increase. contrasting in a direct variation, the ratio in each data is not equivalent. The product of the value in each is equal.

Listed below is one example on Inverse Variation:

Example: If A varies inversely as B and A=2 when B=10,find A when B=4.

Solution:
Given, A=2,B=10 and A∝1/B or,A=k.1/B where k= constant of variation.
So,2=k.1/10 or k=20.

Now again,

A=k.1/B or,A=20.1/4

or,A=5.

Inverse Variation Relationship:

A inverse variation relationship in which one variable increases in value while the other decreases in value is known as an inverse variation relationship.

Consider a trip of 240 miles. The table is the time required for the trip based on the rate of travel.

Listed below is one example on Inverse Variation Relationship:

Example: If y is inversely proportional to x and y = 15 when x = 5. Find the value of y, when x = 25?

Solution:

Let, k = x/y

Plug x = 5 and y = 15 in the above equation

k = 15/5

k = 3

Now the equation becomes 3 = x/y

Now, plug x = 25

3 = y/25

3 * 25 = y

So, y = 75 when x = 25

Tangent

The line meeting the another line at the common point is called tangents. The trigonometric function of an acute angle in an right angled triangle that is the ratio of the length of the opposite side,and the ratio of the length of the adjacent side. A straight line or plane that touches a curved surface at a point but it does not intersect at that point.

Defining Properties of Tangent

Following are the Properties of Tangent:

1. Tangent in terms of sine and cosine:

Since the two triangles ADE and ABC are similar, then

ED / AE = CB / AC.

2. Tangents and right triangles:

Just as the sine and cosine can be found as ratios of sides of the right triangles, so the tangent will use three relations . First is tan A=sin A / cos A. Second is sinA=a/c. Third is cos A=b/c. Dividing a/c by b/c and canceling the cA=a/b. That means is that appear, we conclude that tan the tangent is the opposite divided by the adjacent .

3. A tangent touches a circle exactly one place :

Since a tangent touches the circle at exactly one point, and that point must be perpendicular to a radius.

4. The tangent intersects the circle radius at a 90° angle

It is only when the line is tangent to the circle that the radius will touch that line exactly one point and at this point the 'tangent' must intersect with radius at a 90° angle.