The for binomial theorem is For example The coefficient

 

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

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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 =