(I): The Binomial Probability Distribution:



7.1. The Binomial Probability Distribution

Example:

[pic] representing the number of heads as flipping a fair coin twice.

[pic].

□ □

[pic] T T [pic] (1 combination)

□ □

[pic] H T

T H [pic] (2 combinations)

□ □

[pic] H H [pic] (1 combination)

[pic]

, [pic]

[pic] representing the number of heads as flipping a fair coin 3 times.

□ □ □

[pic] T T T [pic] (1 combination)

□ □ □

H T T

[pic] T H T [pic] (3 combinations)

T T H

□ □ □

H H T

[pic] H T H [pic] (3 combinations)

T H H

□ □ □

[pic] H H H [pic] (1 combination)

[pic]

, [pic]

[pic] representing the number of heads as flipping a fair coin n times.

Then,

n

□ □ …………… □

[pic] T T…………….T [pic]

(1 combination)

n

□ □ …………… □

H T……..……..T

[pic] n T H……… T [pic]

[pic] [pic] [pic] [pic] (n combinations)

T T …………..H

[pic]

[pic]

[pic]

Note: the number of combinations is equivalent to the number of ways as drawing i balls (heads) from n balls (n flips).

Example:

[pic] representing the number of successes over 3 trials.

[pic]

Suppose the probability of the success is [pic] while the probability of failure is [pic].

Then,

□ □ □

[pic] F F F [pic]

(1 combination)

□ □ □

S F F

[pic] F S F [pic]

(3 combinations)

F F S

□ □ □

S S F

[pic] S F S [pic]

(3 combinations)

F S S

□ □ □

[pic] S S S [pic]

(1 combination)

[pic]

, [pic]

[pic] representing the number of successes over n trials.

Then,

n

□ □ …………… □

[pic] F F…………….F [pic]

(1 combination)

n

□ □ ……… □

S F…….. .F

[pic] n F S……… F [pic]

[pic] [pic] [pic] [pic] (n combinations)

F F … .S

[pic]

[pic]

[pic]

From the above example, we readily describe the binomial experiment.

Properties of Binomial Experiment

• X: representing the number of successes over n independent identical trials.

• The probability of a success in a trial is p while the probability of a failure is (1-p).

Binomail Probability Distribution:

Let X be the random variable representing the number of successes of a Binomial experiment. Then, the probability distribution function for X is

[pic].

Properties of Binomial Probability Distribution:

A random variable X has the binomial probability distribution [pic] with parameter [pic], then

[pic]

and

[pic].

[Derivation:]

[pic]

The derivation of [pic] is left as exercise.

How to obtain the binomail probability distribution:

a) Using table of Binomail distribution.

b) Using computer

• by some software, for example, Excel or Minitab.

Online Exercise:

Exercise 7.1.1

Exercise 7.1.2

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