The binomial theorem is a way of expanding a

binomial expression that has been raised to a power that is usually large. The

visual representation of binomial coefficient is the Pascal’s triangle. The

rows of the triangle are enumerated conventionally starting from the first row

which is . It is constructed to allow row zero to only have one

element which is 1. Elements(s) in subsequent row can be found by adding the

number above to the left and the number above to the right. If there is no

number above to the right or left, replace with a zero. Diagram 1.5 will

represent the coefficient of by the 5th

row.

The formula for binomial theorem is

For example

The coefficient is the same as the 3th row of the Pascal’s

triangle

A binomial experiment has these properties

·

It consist of trials repeated n times

·

The trial can have two outcomes. One of the

outcomes is success and the other one is a failure. In this case, success is

the server winning a point and failure is the receiver winning a point

·

Probability of success (p) is constant throughout at 0.55

·

The trial is independent. The outcome of one

trial does not effect other trials

Binomial theorem can be applied in this case because in a

tennis game, there can be two outcomes, either the receiver wins a point or the

server wins a point. The probability of the server on a single trail is given

by w. The probability of failure is .

Lastly there are

n independent and

identical trials.

Successes

Score

Probability

0

Game

to receiver

1

(15,40)

2

(30,30)

3

(40,15)

4

Game

to server

Table 1.2 below will show the probability of a game of tennis

from binomial theorem with n=4. The probability of server winning the game to 0

is .

The game must last four points and all the four points must be acquired by the

server. The server wins the game 0 is given by probability .

Suppose there are 5 points played, it could be one success by

Table 1.2, showing probability

from binomial theorem with n = 4

the server winning only one point and server winning 4

points, or two successes by server winning 2 points and receiver winning 3

points for score of (30, 40), 3 successes by the server winning 3 points and

the receiver winning 2 points (40,30), lastly four successes by server winning

4 points and winning the game and receiver winning 1 point.

If the server wins to 15, the game must last 5 points, the

server must win 4 points and the receiver win 1 point. In this situation, 5

objects be chosen at a time minus number of ways 4 objects chosen 4 at a time

(that i. s server winning to 0), it happens at =

4 ways. Thus the probability of the server winning to 15 is given by .

The server wins to 30, there must be at least six points, the

server must win 4 points and the receiver wins 2 points. To get the ways or

steps, = 10 ways. The probability the server wins to

30 is =

.

Lastly, the probability to reach deuce, server must wins three points out six

points and it happens at ways. Probability of reaching deuce is =