Binomial Distribution

Binomial Distribution

Binomial distribution is the discrete probability distribution of the number of successes in a sequence of n independent yes/no experiments, each of which yields success with probability p. Such a success/failure experiment is also called a Bernoulli experiment or Bernoulli trial; when n = 1, the binomial distribution is a Bernoulli distribution. The binomial distribution is the basis for the popular binomial test of statistical significance.

·       Bernoulli distribution is the same as binomial distribution; it was named after the Swiss mathematician Jakob Bernoulli.

Binomial Experiment- a probability experiment that satisfies the following four requirements:

1.        Each trial can have only two outcomes or outcomes that can be reduced to to two outcomes.

2.      There must be a fixed no of trials.

3.      The outcomes of each trial must be independent of each other.

4.      The probability of a success must be remain the same for each trial.

Notation for thee Binomial Distribution

            P (S) The symbol for the probability of success

            P (F) The symbol for the probability of failure

            p       The numerical probability of success

            q       The numerical probability of failure

            n       The number of trials

            X       The number of successes

Binomial Probability Formula

           

The binomial Distribution has the following properties:

• The mean of distribution-   μ  =n · p 
• The Variance- σ² = n ∙ p ∙ q
• The Standard Deviation-  

Examples:

Suppose a die is tossed 5 times. What is the probability of getting exactly 2 fours?

Solution: This is a binomial experiment in which the number of trials is equal to 5, the number of successes is equal to 2, and the probability of success on a single trial is 1/6 or about 0.167. Therefore, the binomial probability is:

b(2; 5, 0.167) = ₅C₂ * (0.167)² * (0.833)³
b(2; 5, 0.167) = 0.161

Application of Binomial distribution in our daily life;

            Some of my classmates always ask, “andat ginatun-an gd ra ang mga distribution”? magamit ta ra sa kabuhi  ta hw? Anu bay ra pulos na kon hindi related sa math ang kurso nga bul-on namun?’

           

In business;

·       In the insurance field;

Group insurance, which gets cheaper as the group gets larger, is an example of the principle in application; actuarial abnormalities have less influence on total claims.

·       In the auto industry;

when you are  evaluating  the  no. of  cars  are  poorly  painted  on the  system.
The Poisson distribution applies when you are counting the number of objects in a certain volume or the number of events in a certain time period. You know the average number of counts, and wish to know the chance of actually observing various numbers of objects or events. 

·       In the call center management;

to monitor  the  no. of  calls  received / no.  of  telephone machines  breakdown.

·       In the traffic management at a signal;

no of  traffic  flow /  no.  of   traffic  accidents.

·       IN  THE  BUSINESS  OPERATION   FIELD
Applying the technique to calculate, for example, losses from business disruption is, in principle, very straightforward:
* Identify distributions for both the frequency and severity of losses;
* Generate a random value from the frequency distribution to represent the number of losses in a given period;
* Generate random values form the severity distribution for each loss and aggregate to give a total loss for the period;
* Repeat many (several thousand) times and plot an overall loss distribution.
First of all they examine how many outages they suffer per year and it turns out that the mean number of outages in a year is five. They then look at the cost of these outages including:
* Overtime to catch up with lost work;
* Lost customer orders; 

   * Mis-processed customer orders; and
   * Delays / mistakes in billing customers. 

In my own understanding, we can apply binomial probability:

·       WHEN CHOOSING A BUSSINESS TO RUN

For example; you want to run a business of computer shops and a survey from National Cyber Community found that 95% of students go to computer shops every day. If 5 students are selected at random, find the probability that all five will go to computer shops every day. Does your business will succeed?

 

Yes, your business will succeed because more than ½ of the students go to computer shops every day.  

·       WHEN TAKING AN EXAMINATION

For example; if you randomly guesses a 10 multiple-choice questions in an examination. Find the probability that you will get a score of at least 8 correct answers or pass the examination. Each question has five possible choices. Will you continue to randomly guessing? 

With these result I will not continue randomly guessing because the probability that I will pass the 10 multiple-choice examinations with random guessing is 0.0000779264.

(di ko sure kon sakto answer koh)

By: Tito Nuevacobita