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This module describes the characteristics of a binomial experiment and the binomial probability distribution function.

The characteristics of a binomial experiment are:

  1. There are a fixed number of trials. Think of trials as repetitions of an experiment. The letter n denotes the number of trials.
  2. There are only 2 possible outcomes, called "success" and, "failure" for each trial. The letter p denotes the probability of a success on one trial and q denotes the probability of a failure on one trial. p + q = 1 .
  3. The n trials are independent and are repeated using identical conditions. Because the n trials are independent, the outcome of one trial does not help in predicting the outcome of another trial. Another way of saying this is that for each individual trial, the probability, p , of a success and probability, q , of a failure remain the same. For example, randomly guessing at a true - false statistics question has onlytwo outcomes. If a success is guessing correctly, then a failure is guessing incorrectly. Suppose Joe always guesses correctly on any statistics true - false question with probability p 0.6 . Then, q 0.4 .This means that for every true - false statistics question Joe answers, his probability of success ( p 0.6 ) and his probability of failure ( q 0.4 ) remain the same.

The outcomes of a binomial experiment fit a binomial probability distribution . The random variable X = the number of successes obtained in the n independent trials.

The mean, μ , and variance, σ 2 , for the binomial probability distribution is μ np and σ 2 npq . The standard deviation, σ , is then σ npq .

Any experiment that has characteristics 2 and 3 and where n = 1 is called a Bernoulli Trial (named after Jacob Bernoulli who, in the late 1600s, studied them extensively). A binomial experimenttakes place when the number of successes is counted in one or more Bernoulli Trials.

At ABC College, the withdrawal rate from an elementary physics course is 30% for any given term. This implies that, for any given term, 70% of the students stay in theclass for the entire term. A "success" could be defined as an individual who withdrew. The random variable is X = the number of students who withdraw from the randomly selected elementary physics class.

Suppose you play a game that you can only either win or lose. The probability that you win any game is 55% and the probability that you lose is 45%. Each game you play is independent. If you play the game 20times, what is the probability that you win 15 of the 20 games? Here, if you define X = the number of wins, then X takes on the values 0, 1, 2, 3, ..., 20. The probability of a successis p 0.55 . The probability of a failure is q 0.45 . The number of trials is n 20 . The probability question can be stated mathematically as P ( x 15 ) .

A fair coin is flipped 15 times. Each flip is independent. What is the probability of getting more than 10 heads?Let X = the number of heads in 15 flips of the fair coin. X takes on the values 0, 1, 2, 3, ..., 15. Since the coin is fair, p = 0.5 and q = 0.5. The number of trials is n = 15. The probability question can be stated mathematically as P ( x 10 ) .

Approximately 70% of statistics students do their homework in time for it to be collected and graded. Each student does homework independently. In a statistics class of 50 students, what is the probability that at least 40will do their homework on time? Students are selected randomly.

This is a binomial problem because there is only a success or a __________, there are a definite number of trials, and the probability of a success is 0.70 for each trial.


If we are interested in the number of students who do their homework, then how do we define X ?

X = the number of statistics students who do their homework on time

What values does x take on?

0, 1, 2, …, 50

What is a "failure", in words?

Failure is a student who does not do his or her homework on time.

The probability of a success is p = 0.70. The number of trial is n = 50.

If p + q 1 , then what is q ?

q = 0.30

The words "at least" translate as what kind of inequality for the probability question P ( x ____ 40 ) .

greater than or equal to (≥)

The probability question is P ( x 40 ) .

Notation for the binomial: b = binomial probability distribution function

X ~ B ( n , p )

Read this as " X is a random variable with a binomial distribution." The parameters are n and p . n = number of trials p = probability of a success on each trial

It has been stated that about 41% of adult workers have a high school diploma but do not pursue any further education. If 20 adult workers are randomlyselected, find the probability that at most 12 of them have a high school diploma but do not pursue any further education. How many adult workers do you expect to have a high schooldiploma but do not pursue any further education?

Let X = the number of workers who have a high school diploma but do not pursue any further education.

X takes on the values 0, 1, 2, ..., 20 where n = 20 and p = 0.41. q = 1 - 0.41 = 0.59. X ~ B ( 20 , 0.41 )

Find P ( x 12 ) . P ( x 12 ) 0.9738 . (calculator or computer)

Using the TI-83+ or the TI-84 calculators, the calculations are as follows. Go into 2nd DISTR. The syntax for the instructions are

To calculate ( x = value): binompdf( n , p , number) If "number" is left out, the result is the binomial probability table.

To calculate P ( x value ) : binomcdf( n , p , number) If "number" is left out, the result is the cumulative binomial probability table.

For this problem: After you are in 2nd DISTR, arrow down to binomcdf. Press ENTER. Enter20,.41,12). The result is P ( x 12 ) 0.9738 .

If you want to find P ( x 12 ) , use the pdf (binompdf).If you want to find P(x>12) , use 1 - binomcdf(20,.41,12).

The probability at most 12 workers have a high school diploma but do not pursue any further education is 0.9738

The graph of x ~ B ( 20 , 0.41 ) is:

The binomial probability distribution function graph is made up of bars that are fairly normally distributed with an x-axis of 0-20 and a y-axis of 0-0.2 in increments of 0.05.

The y-axis contains the probability of x , where X = the number of workers who have only ahigh school diploma.

The number of adult workers that you expect to have a high school diploma but not pursue any further education is the mean, μ = np = ( 20 ) ( 0.41 ) = 8.2 .

The formula for the variance is σ 2 npq . The standard deviation is σ npq . σ ( 20 ) ( 0.41 ) ( 0.59 ) 2.20 .

The following example illustrates a problem that is not binomial. It violates the condition of independence. ABC College has a student advisorycommittee made up of 10 staff members and 6 students. The committee wishes to choose a chairperson and a recorder. What is the probability that thechairperson and recorder are both students? All names of the committee are put into a box and two names are drawn without replacement . The first name drawn determines the chairperson and the second name the recorder. There aretwo trials. However, the trials are not independent because the outcome of the first trial affects the outcome of the second trial. The probability of a student onthe first draw is 6 16 . The probability of a student on the second draw is 5 15 , when the first draw produces a student. The probability is 6 15 when the first draw produces a staff member. The probability of drawing a student's namechanges for each of the trials and, therefore, violates the condition of independence.

Questions & Answers

Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
why we need to study biomolecules, molecular biology in nanotechnology?
Adin Reply
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
what school?
biomolecules are e building blocks of every organics and inorganic materials.
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
sciencedirect big data base
Introduction about quantum dots in nanotechnology
Praveena Reply
what does nano mean?
Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
Damian Reply
absolutely yes
how to know photocatalytic properties of tio2 nanoparticles...what to do now
Akash Reply
it is a goid question and i want to know the answer as well
characteristics of micro business
for teaching engĺish at school how nano technology help us
Do somebody tell me a best nano engineering book for beginners?
s. Reply
there is no specific books for beginners but there is book called principle of nanotechnology
what is fullerene does it is used to make bukky balls
Devang Reply
are you nano engineer ?
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
what is the actual application of fullerenes nowadays?
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
what is the Synthesis, properties,and applications of carbon nano chemistry
Abhijith Reply
Mostly, they use nano carbon for electronics and for materials to be strengthened.
is Bucky paper clear?
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
so some one know about replacing silicon atom with phosphorous in semiconductors device?
s. Reply
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
Do you know which machine is used to that process?
how to fabricate graphene ink ?
for screen printed electrodes ?
What is lattice structure?
s. Reply
of graphene you mean?
or in general
in general
Graphene has a hexagonal structure
On having this app for quite a bit time, Haven't realised there's a chat room in it.
what is biological synthesis of nanoparticles
Sanket Reply
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
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Source:  OpenStax, Collaborative statistics using spreadsheets. OpenStax CNX. Jan 05, 2016 Download for free at http://legacy.cnx.org/content/col11521/1.23
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