# 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)

what is permutation
how to construct a histogram
I really appreciate that
I want to test linear regression data such as maintenance fees vs house size. Can I use R square, F test to test the relationship? Is the good condition of R square greater than 0.5
yes of course must have use f test and also use t test individually multple coefficients
rishi
Alright
umar
hi frnd I'm akeem by name, I wanna study economics and statistics wat ar d thing I must do to b a great economist
akeem
Is R square cannot analysis linear regression of X vs Y relationship?
Mok
To be an economist you have to be professional in maths
umar
hi frnds
Shehu
what is random sampling what is sample error
@Nistha Kashyap Random sampling is the selection of random items (or random numbers) from the group. A sample error occurs when the selected samples do not truely represent the whole group. The can happen when most or all of the selected samples are taken from only one section of the group;
Ron
Thus the sample is not truely random.
Ron
What is zero sum game?
A game in which there is no profit & no loss to any of the both player.
Milan
Differences between sample mean & population mean
***keydifferences.com/difference-between-sample-mean-and-population-mean.html
Lucien
Not difference in the formula except the notation, sample mean is denoted by x bar and population mean is denoted by mu symbol. There is formula as well as notation between difference variance and standard deviations
Akash
Likely the difference would be in the result, unless the sample is an exact representation of the population (which is unlikely.)
Ron
what is data
Nii
Nii Avin - Data is just a simple way to refer to the numbers in the population, or in the sample used in your calculations.
Ron
what are the types of data
Nii
Data is the very pale android from the Star Trek Enterprise
Andrew
Am Emmanuel from Nigeria
Emmanuel
Am Qudus from Nigeria
Rasak
am Handson from Cameroon
Handson
what is a mode?
Handson
Nii - data is whatever you are sampling. Such as the number of students in each classroom.
Ron
Handson Ndintek - the mode is the number appearing most frequently. Example: 7 9 11 7 4 6 3 7 2. 7 is the mode. In a group such as 7 9 1 4 6 3, there is no mode because no number appears more often than any other.
Ron
hi I want to know how to find class boundary
Baalisi
give me the two types of data
qualitative and quantitative
phoenix
primary and secondary data
Peace
qualitative and quantitative
Prince
Using Cauchy Schwartz inequality,or prove that b2-b1-1=0
what is the ongoing probability that President Trump will remain in the position he has chosen as his viability of his cabinet as he runs for reelection in the primaries of 2020 election year
Terry
what is statistic?
it's a science of collection, organization, analysis and summarizing data to get useful information to make several types of conclusions.which can be used in real life.
anshika
what is the statistical probability that president Trump will remain in the white house after the election of 2020?
Terry
i agree with anshika is right but let me add that such decisions are made in face of uncertainty
Maureen
yes
Stephen
classification of statistic
statistic can classified into many types eassy to understand future values effect
Narendra
what is mean?
Jhasaketan
average value
Narendra
İ want to understand what is t test or neyma. Pearson test ans difference
Yasin
to test the hypotesis ho follws h1 l1/lo
Narendra
Hope this helps. There are three main types of averages. *mean -> average -> (X1+X2+X3+...+Xn) / n *mode -> the element within a set which occurs most. {3,4,5,8,12,3,4,3,3,56} mode = 3 *median -  {3,3,4,5,8,12,56} median = 5 OR {3,4,5,8,12,56} median = 6.5
Jack
conceptual approach to limits
how are limits derived?
lameck
an entire section of calculus is devoted to that explanation.
Pitior
what is statistics?
statistics :- can be defined as the branches of mathematics that deals with the summarizing, analysing,organization and interpretation of data.
Usman
well said
Venkat
can we find Z value on calculator with out using Z table
no
Pitior
why
Maham
can another way is possible ?
Maham
Well you could make a table. And as the function you use the one used at the z table
Luca
The normal function is only one way, so you can only try using different numbers until you get the probability that you have. So that is easier if you have a table
Luca
me don't know nothing about z table and don't know how to see the z value on table can you tell me please how see the value on table
Maham
The z table is the table of the standard normal distribution
Luca
You can look it up on internet, its easier than writing down the normal distribution function (with an integral) and doing a table in the calculator
Luca
OK thanks luca
Maham
yes use pnorm in r
Venkat
pnorm(2.3,mean=0,sd=1)
Venkat
pnorm?
Maham
do u have r software
Venkat
no
Maham
its with tht u will get
Venkat
Venkat
z mathportal calculator
Venkat
calculator
Venkat
OK venkat thanks
Maham
welcome
Venkat
have calculator but don't know how find z value
Maham
ti83
Venkat
hey guys I'm from computer background so what are the concepts I supposed to prepare for interview in statistics
Alwin
descriptive stats
Venkat
inferential stats
Venkat
outlier treatment
Venkat
boxplot
Venkat
ok
Alwin
assumption of linear regression
Venkat
logistic regression
Venkat
k means clustering
Venkat
exact syllabus?
Alwin
type. analytics vidya interview questions statistics
Venkat
listen. data also
Venkat
like this forum
Jameel
My question is "is it only stats?"
Jameel
wer is the problem
Venkat
how find straight line equation in regression
u can find using excel
Venkat
or r studio
Venkat
for regression
Venkat
shall i help
Venkat
im an expert
Venkat
by giving a value to x,y
Ibrokhim
first provide data
Venkat
ill solve and guve
Venkat
ive
Venkat
Maham
maham you posted data
Venkat
Venkat
ok
Maham
x:1,2,3,4,5 y:2,5,6,8,9
Maham
regredsion equation is
Venkat
y=0.9+1.7x
Venkat
reg eq is y=0.9+1.7x
Venkat
slope = 1.7
Venkat
yintercept = 0.9
Venkat
Venkat
thanx venkat naveen😊
Maham
welcome
Venkat