# 4.3 Binomial distribution

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There are three characteristics of a binomial experiment.

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 two 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 only two 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 are μ = np and σ 2 = npq . The standard deviation, σ , is then σ = $\sqrt{npq}$ .

Any experiment that has characteristics two and three and where n = 1 is called a Bernoulli Trial (named after Jacob Bernoulli who, in the late 1600s, studied them extensively). A binomial experiment takes 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 the class for the entire term. A "success" could be defined as an individual who withdrew. The random variable X = the number of students who withdraw from the randomly selected elementary physics class.

## Try it

The state health board is concerned about the amount of fruit available in school lunches. Forty-eight percent of schools in the state offer fruit in their lunches every day. This implies that 52% do not. What would a "success" be in this case?

a school that offers fruit in their lunch every day

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 20 times, write the function that describes the probability that you win 15 of the 20 times. Here, if you define X as the number of wins, then X takes on the values 0, 1, 2, 3, ..., 20. The probability of a success is 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).

## Try it

A trainer is teaching a dolphin to do tricks. The probability that the dolphin successfully performs the trick is 35%, and the probability that the dolphin does not successfully perform the trick is 65%. Out of 20 attempts, you want to find the probability that the dolphin succeeds 12 times. State the probability question mathematically.

P ( x = 12)

#### Questions & Answers

According to my understanding, statistics are numerical data or information presented in charts or graphs
Kristen Reply
thanks
Said
what is statistics
Elisha Reply
how to find critical angle
Muhammad Reply
what is statistics ?
Samad
Statistics is deal with the collection, analysis, interpretation, presentation of data to drawn out conclusions.
Ajao
collection presentation analysis interpretation Kia he sir
Samad
In my own perspective statistics is deal with the collection, analysis, interpretation, presentation of data before will drawn our conclusions
Ajao
thank you sir pragaph bejo
Samad
In simple way it is the technique of problem solving by providing probabilistic answer to questions otherwise whose exact answer is not possible..besides it is the support to every subject you study.
being
data
Deng
Data is information that has been translated into a form that is efficient for movement or processing
Mansour
group the scores below into a class interval of seven and obtain the following: 1.. i..class interval ii..tally iii..frequency iv..mid point v...cummulative frequency less than vi..cummulative frequency more than vii..relative frequency viii...relative percentage
Icecool
2..draw a histogram and frequency polygon to sick it the statistical table
Icecool
20,30,39,25,50,26,39,32,43,20,30,40,50,10,16,12,19,31,32,26,29,27,28,42,63,48,11,26,35,36,37,38,39,39,35,29,53,37,32,24,36,43,48,42,51,37,39,39,24,7,8,13,62,52,69,68,37,38,39,40,41,39,54
Icecool
2.draw a histogram and frequency polygon to sick it the statistical
Deng
2.draw a histogram and frequency polygon to dipict the statistical table
Icecool
Describe the organizational setup of bureau of statics government of sindh give its past achievement in brief
Awais Reply
Life Table and its construction
Awais
only asking what is hospital statistics
Suly Reply
What's the meaning Of statistic
Farouk Reply
what's the meaning of statistic
Musa
The mean is the average of the numbers. It is easy to calculate: add up all the numbers, then divide by how many numbers there are. In other words it is the sum divided by the count.
khama Reply
pls I want to know the characterize of mean
Trixie Reply
I want solution to; x,x+1,3.5,3.4 and 2.5 is 3.1. find the value of x
Trixie
mean is nothing but average.
NAMDEV
(x+x+1+3.5+3.4+2.5)/5=3.1 2x+10.4=3.1*5 x=(15.5-10.4)/2 x=2.55
NAMDEV
How. Pls explain or break it down thanks
Icecool
the sum of all five divided by the 5 which is the average is equal to 3.1 so you sum them, divide and equate to 3.1. then solve for X by making X the subject
Pascale
Think of it as your grade, or gpa. how do you find the average of it?
La
great
Rushikesh
nicely explained
Rushikesh
What is the meaning of statistics and it's relevance in the social sciences
Icecool Reply
statistics is the branch of mathematics that using math& statistics theories to interpret data to information which is supporting the decision makers in social sciences and applied too. statistics is decision making science now aday
Ishag
I want to slove this math
Enny Reply
I'm not seeing the Math problem?
Debra
the data below where from 1 miles. miles 1:8380, 8210,3836,7840,7910,7510,7720,7750,8100,7690 test wherever their difference is significance at 5% level with population of #8000.
Enny
1*1
Enny
me too
New
enny i don't see any math problems
New
hello my friend. how can you help me to solve math
ott
help me the interquartile range and semi interquartile range of grouped data
Ishraaq
what is the difference between macro and micro data?
Sani Reply
Macrodata consist of 'headcounts', Microdata contain data on individuals,
Cherry
macro is a big business and micro is small business
Mukhtar
macro means whole or big and micro means part or small therefore in relation to data, macro data will be big or whole data and micro data will be small or individual part of a data. for example taking the whole data of an organisation or economy that will be macro data and taking the data on employe
Pascale
employees will be micro data
Pascale
How to determine appropriate statistics tool for my data
Jayr Reply
hi
Juliana
hi
Emy
You cannot know everything! Statistics is a vast subject so don’t be afraid to say that you don’t know. Ask others for help or look up information on the internet to help. Don’t assume that the student knows anything about the technique they are suggesting!
Kalu
well
Tahir
my class starts at February 4th
Elaine
P is the probability that a given die shows even number. To test H0 : P =1/2 against H1 : P=1/3, following procedure is adopted. Toss the die twice and accept H0 if both times it shows even number. Find the probabilities of type I and type II errors
Loyolite Reply
what is statistics?
bilal Reply
what is data
natnael
It Is the collection, collation,Analysing Interpretation,and presentation of data....
Tosayo
statistics is a science that studies numerical values assigned to objects, attributes or variables in order to characterize and make the variables more meaningful.
Okosa
statistics is the science of collection, analysis, Interpretation & presentation of numerical data
Kishan
Numeric data
Tosayo
if you think about normal number and operations of math such as fractions and percent values, statisitics is when it comes down to it a lot of just that - counting and then fractions, then that data and those counts (and ratios)/fractions that were used or counted in one group will represent a pop.
Jesus
An example is, lets say you are in charge of making large orders of supplies and that based on your need of X bricks = X feet of building - now thats the math, well stats is using that math in a way the next person making orders can base those orders of what they will need based on saying "in this
Jesus
..case, when Bilal needed 75 feet of building walls that 150 bricks were used" - now, that helps to get one job done but if we then watch those jobs one at a time, and count that 5/5 times the count was 150 bricks total - the new person could order 150 bricks with a high probability of being right
Jesus
If X is a Uniform random variable in [ -2, 2 ], find the pdf of Y X  and E Y[ ].
Kezang Reply
could you resend the question it is not clear after " the PDF of---" ?
Anji
ap kis year ki student ho
Shahjee
Mastrs ist year
Anaya
I'm from 2nd year BHU
abhishek
ap kis year ki student ho
Shahjee
I'm student of bs(hons) 1st year
bilal
tell me good definition of statistics?
bilal

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Source:  OpenStax, Introductory statistics. OpenStax CNX. May 06, 2016 Download for free at http://legacy.cnx.org/content/col11562/1.18
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